MAT2305 LINEAR ALGEBRA :Week2 · Outline for Week 2 Chapter 2 Matrices 2.1 Matrices and Matrix...

MAT2305 LINEAR ALGEBRA :Week2

Thanatyod Jampawai, Ph.D.

Thanatyod Jampawai, Ph.D. MAT2305 LINEAR ALGEBRA :Week2 1 / 60

Review Week 1

Linear Equation and Linear System of Linear EquationsSolutionConsistent and InconsistentGuass EliminationAugmented MatrixElementary Row Operations (EROs)Row Echelon Form (REF) vs Row Reduced Echelon Form (RREF)Guass-Jordan Elimination

Outline for Week 2

Chapter 2 Matrices2.1 Matrices and Matrix Operations2.2 Inverse Matrix2.3 Linear System with Invertibility

Assignment 2

Matrices

Chapter 2 Matrices

Matrices Matrices and Matrix Operations

2.1 Matrices and Matrix Operations

Aj.Thanatyod Jampwai, faculty of education, SSRU, spent lecture two sujectsduring a certain week for semester 2/2017:

Mon. Tues. Wed. Thurs. Fri. Sat. SunMAP1405: Number Theory 0 6 0 0 0 0 0MAT2305: Linear Algebra 6 0 0 0 0 0 0

The following rectangular array of number with 2 rows and 7 columns iscalled a matrix: [

0 6 0 0 0 0 06 0 0 0 0 0 0

Definition (2.1.1)

A matrix is a rectangular array of numbers. The numbers in the array are

called the entries in the matrix.

0 6 0 0 0 0 06 0 0 0 0 0 0

Definition (2.1.1)

0 6 0 0 0 0 06 0 0 0 0 0 0

Definition (2.1.1)

The size of a matrix is described in term of the number of rows andcolumns it contains.

[−1 3 0 21 0 1 0

]is a martix that its size is 2 by 4 (written 2× 4).

A matrix with only one column is called a column matrix (or a columnvector),

row matrix:[1 0 2 3 5

]A matrix with only one row is called a row matrix (or a row vector).

column matrix:[12

The size of a matrix is described in term of the number of rows andcolumns it contains. [

−1 3 0 21 0 1 0

column matrix:[12

−1 3 0 21 0 1 0

A matrix with only one row is called a row matrix (or a row vector).

column matrix:[12

−1 3 0 21 0 1 0

column matrix:[12

A general m× n matrix as

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

The preceding matrix A can be written as

[aij ]m×n or [aij ].

The entry that occurs in row i and column j of a matrix A will be denoted by aij .

A = [aij ]3×5 =

a11 a12 a13 a14 a15a21 a22 a23 a24 a25a31 a32 a33 a34 a35

A matrix A with n rows and n columns is called a square matrix of order n , and entries

a11, a22, ..., ann are said to be on the main diagonal of A.

A = [aij ]3×3 =

a11 a12 a13a21 a22 a23a31 a32 a33

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

A = [aij ]3×5 =

A = [aij ]3×3 =

a11 a12 a13a21 a22 a23a31 a32 a33

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

A = [aij ]3×5 =

A = [aij ]3×3 =

a11 a12 a13a21 a22 a23a31 a32 a33

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

A = [aij ]3×5 =

A = [aij ]3×3 =

a11 a12 a13a21 a22 a23a31 a32 a33

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

A = [aij ]3×5 =

A matrix A with n rows and n columns is called a square matrix of order n ,

and entries

A = [aij ]3×3 =

a11 a12 a13a21 a22 a23a31 a32 a33

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

A = [aij ]3×5 =

A = [aij ]3×3 =

a11 a12 a13a21 a22 a23a31 a32 a33

Definition (2.1.2)

Two matrices are defined to be equal if they have the same size and thiercorresponding entries are eqaul.

Example (2.1.1)

Compute x and y if A = B.

[1 x+ 13 2

]and B =

[1 y + 2

1− y 2

Solution

x+ 1 = y + 2 → x− y = 1

3 = 1− y → y = −2 ∴ x = −1

Therefore, x = −1 and y = −2 #

Definition (2.1.2)

Example (2.1.1)

[1 x+ 13 2

]and B =

[1 y + 2

1− y 2

Solution

x+ 1 = y + 2 → x− y = 1

3 = 1− y → y = −2 ∴ x = −1

Definition (2.1.2)

Example (2.1.1)

[1 x+ 13 2

]and B =

[1 y + 2

1− y 2

Solution

x+ 1 = y + 2 → x− y = 1

3 = 1− y → y = −2 ∴ x = −1

Sum and Difference of Matrices

Definition (2.1.3)

Let A = [aij ] and B = [bij ] be matrices of the same size.

The sum of A and B

A+B = [aij + bij ]

The difference of A and B

A−B = [aij − bij ]

Matrices of different sizes cannot be added or substracted.

Example (2.1.2)

Find A+B, A−B, A+ C and B − C.

1 3 43 2 −20 −1 3

1 2 3−4 2 15 1 0

and C =

[1 67 2

Solution

1 3 43 2 −20 −1 3

1 2 3−4 2 15 1 0

2 5 7−1 4 −15 0 3

A−B =

1 3 43 2 −20 −1 3

1 2 3−4 2 15 1 0

0 1 17 0 −3−5 −2 3

A+ C and B − C are undefined because two matrices are different sizes. #

Example (2.1.2)

1 3 43 2 −20 −1 3

1 2 3−4 2 15 1 0

and C =

[1 67 2

Solution

1 3 43 2 −20 −1 3

1 2 3−4 2 15 1 0

2 5 7−1 4 −15 0 3

A−B =

1 3 43 2 −20 −1 3

1 2 3−4 2 15 1 0

0 1 17 0 −3−5 −2 3

Example (2.1.2)

1 3 43 2 −20 −1 3

1 2 3−4 2 15 1 0

and C =

[1 67 2

Solution

1 3 43 2 −20 −1 3

1 2 3−4 2 15 1 0

2 5 7−1 4 −15 0 3

A−B =

1 3 43 2 −20 −1 3

1 2 3−4 2 15 1 0

0 1 17 0 −3−5 −2 3

Scalar Multiplication

Definition (2.1.4)

Let A = [aij ] be any matrix and c be any scalar. Then the product cA is thematrix obtained by multuplying each entry of the matrix A by c. The matrixcA is said to be a scalar multiple of A .

cA = [caij ]

Example (2.1.3)

For the matrices

[1 −4 4 10 3 −2 10

], B =

[1 6 3 53 2 −1 2

]and C =

[5 0 9 57 2 0 −3

Compute 2A, 3B −A, C − (A+ 2B) and 2(A+B)− 3C

Scalar Multiplication

Definition (2.1.4)

Let A = [aij ] be any matrix and c be any scalar. Then the product cA is thematrix obtained by multuplying each entry of the matrix A by c. The matrixcA is said to be a scalar multiple of A .

cA = [caij ]

Example (2.1.3)

For the matrices

[1 −4 4 10 3 −2 10

], B =

[1 6 3 53 2 −1 2

]and C =

[5 0 9 57 2 0 −3

Compute 2A, 3B −A, C − (A+ 2B) and 2(A+B)− 3C

Solution

2A = 2

[1 −4 4 10 3 −2 10

[2 −8 8 20 6 −4 20

3B −A = 3

[1 6 3 53 2 −1 2

[1 −4 4 10 3 −2 10

[3 18 9 159 6 −3 6

[1 −4 4 10 3 −2 10

[2 22 5 149 3 −1 −4

C − (A+ 2B) =

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[1 6 3 53 2 −1 2

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[2 12 6 106 4 −2 4

[5 0 9 57 2 0 −3

[3 −8 10 116 7 −4 14

[2 8 −1 −61 −5 4 −17

Solution

2A = 2

[1 −4 4 10 3 −2 10

[2 −8 8 20 6 −4 20

3B −A = 3

[1 6 3 53 2 −1 2

[1 −4 4 10 3 −2 10

[3 18 9 159 6 −3 6

[1 −4 4 10 3 −2 10

[2 22 5 149 3 −1 −4

C − (A+ 2B) =

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[1 6 3 53 2 −1 2

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[2 12 6 106 4 −2 4

[5 0 9 57 2 0 −3

[3 −8 10 116 7 −4 14

[2 8 −1 −61 −5 4 −17

Solution

2A = 2

[1 −4 4 10 3 −2 10

[2 −8 8 20 6 −4 20

3B −A = 3

[1 6 3 53 2 −1 2

[1 −4 4 10 3 −2 10

[3 18 9 159 6 −3 6

[1 −4 4 10 3 −2 10

[2 22 5 149 3 −1 −4

C − (A+ 2B) =

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[1 6 3 53 2 −1 2

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[2 12 6 106 4 −2 4

[5 0 9 57 2 0 −3

[3 −8 10 116 7 −4 14

[2 8 −1 −61 −5 4 −17

Solution

2A = 2

[1 −4 4 10 3 −2 10

[2 −8 8 20 6 −4 20

3B −A = 3

[1 6 3 53 2 −1 2

[1 −4 4 10 3 −2 10

[3 18 9 159 6 −3 6

[1 −4 4 10 3 −2 10

[2 22 5 149 3 −1 −4

C − (A+ 2B) =

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[1 6 3 53 2 −1 2

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[2 12 6 106 4 −2 4

[5 0 9 57 2 0 −3

[3 −8 10 116 7 −4 14

[2 8 −1 −61 −5 4 −17

Solution

2A = 2

[1 −4 4 10 3 −2 10

[2 −8 8 20 6 −4 20

3B −A = 3

[1 6 3 53 2 −1 2

[1 −4 4 10 3 −2 10

[3 18 9 159 6 −3 6

[1 −4 4 10 3 −2 10

[2 22 5 149 3 −1 −4

C − (A+ 2B) =

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[1 6 3 53 2 −1 2

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[2 12 6 106 4 −2 4

[5 0 9 57 2 0 −3

[3 −8 10 116 7 −4 14

[2 8 −1 −61 −5 4 −17

Solution

2A = 2

[1 −4 4 10 3 −2 10

[2 −8 8 20 6 −4 20

3B −A = 3

[1 6 3 53 2 −1 2

[1 −4 4 10 3 −2 10

[3 18 9 159 6 −3 6

[1 −4 4 10 3 −2 10

[2 22 5 149 3 −1 −4

C − (A+ 2B) =

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[1 6 3 53 2 −1 2

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[2 12 6 106 4 −2 4

[5 0 9 57 2 0 −3

[3 −8 10 116 7 −4 14

[2 8 −1 −61 −5 4 −17

Solution

2A = 2

[1 −4 4 10 3 −2 10

[2 −8 8 20 6 −4 20

3B −A = 3

[1 6 3 53 2 −1 2

[1 −4 4 10 3 −2 10

[3 18 9 159 6 −3 6

[1 −4 4 10 3 −2 10

[2 22 5 149 3 −1 −4

C − (A+ 2B) =

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[1 6 3 53 2 −1 2

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[2 12 6 106 4 −2 4

[5 0 9 57 2 0 −3

[3 −8 10 116 7 −4 14

[2 8 −1 −61 −5 4 −17

Solution

2A = 2

[1 −4 4 10 3 −2 10

[2 −8 8 20 6 −4 20

3B −A = 3

[1 6 3 53 2 −1 2

[1 −4 4 10 3 −2 10

[3 18 9 159 6 −3 6

[1 −4 4 10 3 −2 10

[2 22 5 149 3 −1 −4

C − (A+ 2B) =

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[1 6 3 53 2 −1 2

[5 0 9 57 2 0 −3

]− (

[1 −4 4 10 3 −2 10

[2 12 6 106 4 −2 4

[5 0 9 57 2 0 −3

[3 −8 10 116 7 −4 14

[2 8 −1 −61 −5 4 −17

Solution

2(A+B)− 3C = 2(

[1 −4 4 10 3 −2 10

[1 6 3 53 2 −1 2

])− 3

[5 0 9 57 2 0 −3

[2 2 7 63 5 −3 12

[15 0 27 1521 6 0 −6

[4 4 14 126 10 −6 24

[15 0 27 1521 6 0 −6

[−11 4 −13 −3−15 4 −6 30

Matrix Multiplication

Definition (2.1.5)

Let A = [aij ] be an m× r matrix and B = [bij ] be an r × n matrix. Then

a11 a12 · · · a1ra21 a22 · · · a2r...

......

ai1 ai2 · · · air...

......

am1 am2 · · · amr

b11 b12 · · · b1j · · · b1nb21 b22 · · · b2j · · · b2n...

......

...br1 br2 · · · brj · · · brn

the entry cij of product AB which

AB = [aij ]m×r[bij ]r×n = [cij ]m×n

is given bycij = ai1b1j + ai2b2j + · · ·+ airbrj .

Example (2.1.4)

Find AB, BA, BC, CB, CA and AC.

[1 3 45 2 6

], B =

1 67 20 1

and C =

0 6 −15 1 20 1 0

Solution

[1 3 45 2 6

] 1 67 20 1

[1 + 21 + 0 6 + 6 + 45 + 14 + 0 30 + 4 + 6

[22 1619 40

1 67 20 1

[ 1 3 45 2 6

1 + 30 3 + 12 4 + 367 + 10 21 + 4 28 + 120 + 5 0 + 2 0 + 6

31 15 4017 25 405 2 6

1 67 20 1

0 6 −15 1 20 1 0

Undefined #

Example (2.1.4)

[1 3 45 2 6

], B =

1 67 20 1

and C =

0 6 −15 1 20 1 0

Solution

[1 3 45 2 6

] 1 67 20 1

[1 + 21 + 0 6 + 6 + 45 + 14 + 0 30 + 4 + 6

[22 1619 40

1 67 20 1

[ 1 3 45 2 6

1 + 30 3 + 12 4 + 367 + 10 21 + 4 28 + 120 + 5 0 + 2 0 + 6

31 15 4017 25 405 2 6

1 67 20 1

0 6 −15 1 20 1 0

Undefined #

Example (2.1.4)

[1 3 45 2 6

], B =

1 67 20 1

and C =

0 6 −15 1 20 1 0

Solution

[1 3 45 2 6

] 1 67 20 1

[1 + 21 + 0 6 + 6 + 45 + 14 + 0 30 + 4 + 6

[22 1619 40

1 67 20 1

[ 1 3 45 2 6

1 + 30 3 + 12 4 + 367 + 10 21 + 4 28 + 120 + 5 0 + 2 0 + 6

31 15 4017 25 405 2 6

1 67 20 1

0 6 −15 1 20 1 0

Undefined #

Example (2.1.4)

[1 3 45 2 6

], B =

1 67 20 1

and C =

0 6 −15 1 20 1 0

Solution

[1 3 45 2 6

] 1 67 20 1

[1 + 21 + 0 6 + 6 + 45 + 14 + 0 30 + 4 + 6

[22 1619 40

1 67 20 1

[ 1 3 45 2 6

1 + 30 3 + 12 4 + 367 + 10 21 + 4 28 + 120 + 5 0 + 2 0 + 6

31 15 4017 25 405 2 6

1 67 20 1

0 6 −15 1 20 1 0

Undefined #

Example (2.1.4)

[1 3 45 2 6

], B =

1 67 20 1

and C =

0 6 −15 1 20 1 0

Solution

[1 3 45 2 6

] 1 67 20 1

[1 + 21 + 0 6 + 6 + 45 + 14 + 0 30 + 4 + 6

[22 1619 40

1 67 20 1

[ 1 3 45 2 6

1 + 30 3 + 12 4 + 367 + 10 21 + 4 28 + 120 + 5 0 + 2 0 + 6

31 15 4017 25 405 2 6

1 67 20 1

0 6 −15 1 20 1 0

Undefined #

Example (2.1.4)

[1 3 45 2 6

], B =

1 67 20 1

and C =

0 6 −15 1 20 1 0

Solution

[1 3 45 2 6

] 1 67 20 1

[1 + 21 + 0 6 + 6 + 45 + 14 + 0 30 + 4 + 6

[22 1619 40

1 67 20 1

[ 1 3 45 2 6

1 + 30 3 + 12 4 + 367 + 10 21 + 4 28 + 120 + 5 0 + 2 0 + 6

31 15 4017 25 405 2 6

1 67 20 1

0 6 −15 1 20 1 0

Undefined #

Example (2.1.4)

[1 3 45 2 6

], B =

1 67 20 1

and C =

0 6 −15 1 20 1 0

Solution

[1 3 45 2 6

] 1 67 20 1

[1 + 21 + 0 6 + 6 + 45 + 14 + 0 30 + 4 + 6

[22 1619 40

1 67 20 1

[ 1 3 45 2 6

1 + 30 3 + 12 4 + 367 + 10 21 + 4 28 + 120 + 5 0 + 2 0 + 6

31 15 4017 25 405 2 6

1 67 20 1

0 6 −15 1 20 1 0

Undefined #

Example (2.1.4)

[1 3 45 2 6

], B =

1 67 20 1

and C =

0 6 −15 1 20 1 0

Solution

[1 3 45 2 6

] 1 67 20 1

[1 + 21 + 0 6 + 6 + 45 + 14 + 0 30 + 4 + 6

[22 1619 40

1 67 20 1

[ 1 3 45 2 6

1 + 30 3 + 12 4 + 367 + 10 21 + 4 28 + 120 + 5 0 + 2 0 + 6

31 15 4017 25 405 2 6

1 67 20 1

0 6 −15 1 20 1 0

Undefined #

Example (2.1.4)

[1 3 45 2 6

], B =

1 67 20 1

and C =

0 6 −15 1 20 1 0

Solution

[1 3 45 2 6

] 1 67 20 1

[1 + 21 + 0 6 + 6 + 45 + 14 + 0 30 + 4 + 6

[22 1619 40

1 67 20 1

[ 1 3 45 2 6

1 + 30 3 + 12 4 + 367 + 10 21 + 4 28 + 120 + 5 0 + 2 0 + 6

31 15 4017 25 405 2 6

1 67 20 1

0 6 −15 1 20 1 0

Undefined #

Solution

0 6 −15 1 20 1 0

1 67 20 1

0 + 42 + 0 0 + 12− 15 + 7 + 0 30 + 2 + 20 + 7 + 0 0 + 2 + 0

42 1112 347 2

0 6 −15 1 20 1 0

[ 1 3 45 2 6

]Undefined #

[1 3 45 2 6

] 0 6 −15 1 20 1 0

[0 + 15 + 0 6 + 3 + 4 −1 + 6 + 00 + 10 + 0 30 + 2 + 6 −5 + 4 + 0

[15 13 510 38 −1

Solution

0 6 −15 1 20 1 0

1 67 20 1

0 + 42 + 0 0 + 12− 15 + 7 + 0 30 + 2 + 20 + 7 + 0 0 + 2 + 0

42 1112 347 2

0 6 −15 1 20 1 0

[ 1 3 45 2 6

]Undefined #

[1 3 45 2 6

] 0 6 −15 1 20 1 0

[0 + 15 + 0 6 + 3 + 4 −1 + 6 + 00 + 10 + 0 30 + 2 + 6 −5 + 4 + 0

[15 13 510 38 −1

Solution

0 6 −15 1 20 1 0

1 67 20 1

0 + 42 + 0 0 + 12− 15 + 7 + 0 30 + 2 + 20 + 7 + 0 0 + 2 + 0

42 1112 347 2

0 6 −15 1 20 1 0

[ 1 3 45 2 6

]Undefined #

[1 3 45 2 6

] 0 6 −15 1 20 1 0

[0 + 15 + 0 6 + 3 + 4 −1 + 6 + 00 + 10 + 0 30 + 2 + 6 −5 + 4 + 0

[15 13 510 38 −1

Solution

0 6 −15 1 20 1 0

1 67 20 1

0 + 42 + 0 0 + 12− 15 + 7 + 0 30 + 2 + 20 + 7 + 0 0 + 2 + 0

42 1112 347 2

0 6 −15 1 20 1 0

[ 1 3 45 2 6

Undefined #

[1 3 45 2 6

] 0 6 −15 1 20 1 0

[0 + 15 + 0 6 + 3 + 4 −1 + 6 + 00 + 10 + 0 30 + 2 + 6 −5 + 4 + 0

[15 13 510 38 −1

Solution

0 6 −15 1 20 1 0

1 67 20 1

0 + 42 + 0 0 + 12− 15 + 7 + 0 30 + 2 + 20 + 7 + 0 0 + 2 + 0

42 1112 347 2

0 6 −15 1 20 1 0

[ 1 3 45 2 6

]Undefined #

[1 3 45 2 6

] 0 6 −15 1 20 1 0

[0 + 15 + 0 6 + 3 + 4 −1 + 6 + 00 + 10 + 0 30 + 2 + 6 −5 + 4 + 0

[15 13 510 38 −1

Solution

0 6 −15 1 20 1 0

1 67 20 1

0 + 42 + 0 0 + 12− 15 + 7 + 0 30 + 2 + 20 + 7 + 0 0 + 2 + 0

42 1112 347 2

0 6 −15 1 20 1 0

[ 1 3 45 2 6

]Undefined #

[1 3 45 2 6

] 0 6 −15 1 20 1 0

[0 + 15 + 0 6 + 3 + 4 −1 + 6 + 00 + 10 + 0 30 + 2 + 6 −5 + 4 + 0

[15 13 510 38 −1

Solution

0 6 −15 1 20 1 0

1 67 20 1

0 + 42 + 0 0 + 12− 15 + 7 + 0 30 + 2 + 20 + 7 + 0 0 + 2 + 0

42 1112 347 2

0 6 −15 1 20 1 0

[ 1 3 45 2 6

]Undefined #

[1 3 45 2 6

] 0 6 −15 1 20 1 0

[0 + 15 + 0 6 + 3 + 4 −1 + 6 + 00 + 10 + 0 30 + 2 + 6 −5 + 4 + 0

[15 13 510 38 −1

Solution

0 6 −15 1 20 1 0

1 67 20 1

0 + 42 + 0 0 + 12− 15 + 7 + 0 30 + 2 + 20 + 7 + 0 0 + 2 + 0

42 1112 347 2

0 6 −15 1 20 1 0

[ 1 3 45 2 6

]Undefined #

[1 3 45 2 6

] 0 6 −15 1 20 1 0

[0 + 15 + 0 6 + 3 + 4 −1 + 6 + 00 + 10 + 0 30 + 2 + 6 −5 + 4 + 0

[15 13 510 38 −1

Theorem (2.1.1)

Assuming that the sizes of the matrices are such that indicated operations can beperformed, the following rules of matrix arithemetic are valid.

1 A+B = B +A (Commutative law for addition)

2 A+ (B + C) = (A+B) + C (Associative law for addition)

3 A(BC) = (AB)C (Commutative law for multiplication)

4 A(B + C) = AB +AC (Left distributive law)

5 (B + C)A = BA+ CA (Right distributive law)

Theorem (2.1.2)

Assuming that the sizes of the matrices are such that indicated operations can beperformed, the following rules of matrix arithemetic are valid. For any scalars a and b,

1 A(B − C) = AB −AC and (B − C)A = BA− CA

2 a(B + C) = aB + aC and a(B − C) = aB − aC

3 (a+ b)C = aC + bC and (a− b)C = aC − bC

4 (ab)C = a(bC) = b(aC) and a(BC) = B(aC) = (aB)C

Theorem (2.1.1)

Assuming that the sizes of the matrices are such that indicated operations can beperformed, the following rules of matrix arithemetic are valid.

1 A+B = B +A (Commutative law for addition)

2 A+ (B + C) = (A+B) + C (Associative law for addition)

3 A(BC) = (AB)C (Commutative law for multiplication)

4 A(B + C) = AB +AC (Left distributive law)

5 (B + C)A = BA+ CA (Right distributive law)

Theorem (2.1.2)

Assuming that the sizes of the matrices are such that indicated operations can beperformed, the following rules of matrix arithemetic are valid. For any scalars a and b,

1 A(B − C) = AB −AC and (B − C)A = BA− CA

2 a(B + C) = aB + aC and a(B − C) = aB − aC

3 (a+ b)C = aC + bC and (a− b)C = aC − bC

4 (ab)C = a(bC) = b(aC) and a(BC) = B(aC) = (aB)C

Transpose

Definition (2.1.6)

Let A = [aij ] be an m× n matrix. Then the traspose of A , denoted by AT ,is defined to be

AT = [aji]n×m

Example (2.1.5)

Find transposes of the following matrices.

[1 5 62 2 7

], B =

[3 95 8

], C =

[1 3 −2

]and D =

Solution

1 25 26 7

, BT =

[3 59 8

], CT =

13−2

and DT =[1 2 3

Transpose

Definition (2.1.6)

AT = [aji]n×m

Example (2.1.5)

[1 5 62 2 7

], B =

[3 95 8

], C =

[1 3 −2

]and D =

Solution

1 25 26 7

, BT =

[3 59 8

], CT =

13−2

and DT =[1 2 3

Transpose

Definition (2.1.6)

AT = [aji]n×m

Example (2.1.5)

[1 5 62 2 7

], B =

[3 95 8

], C =

[1 3 −2

]and D =

Solution

1 25 26 7

, BT =

[3 59 8

], CT =

13−2

and DT =[1 2 3

Theorem (2.1.3)

Assuming that the sizes of the matrices are such that indicated operations canbe performed, the following rules of transpose of matrices are valid.

1 (A+B)T = AT +BT and (A−B)T = AT −BT

2 (AB)T = BTAT

3 (AT )T = A

4 (aA)T = a(AT ) where a is a scalar.

Zero Matrix

Definition (2.1.7)

A zero matrix is a matrix whose entries are zero, denoted by 0.

Theorem (2.1.4)

Assuming that the sizes of the matrices are such that indicated operations canbe performed, the following rules of matrix arithemetic are valid.

1 A+ 0 = A = 0 +A (0 is the identity under addition of matrix)

2 A+ (−A) = 0 = (−A) +A (additive inverse of matrix)

3 0−A = −A

4 0A = 0 = A0

Zero Matrix

Definition (2.1.7)

A zero matrix is a matrix whose entries are zero, denoted by 0.

Theorem (2.1.4)

Assuming that the sizes of the matrices are such that indicated operations canbe performed, the following rules of matrix arithemetic are valid.

1 A+ 0 = A = 0 +A (0 is the identity under addition of matrix)

2 A+ (−A) = 0 = (−A) +A (additive inverse of matrix)

3 0−A = −A

4 0A = 0 = A0

Matrices Inverse Matrix

2.2 Inverse Matrix

Definition (2.2.1)

An identity matrix , denoted by I, is a square matrix with 1’s on the maindoagonal and 0’s off the main diagonal.

If it is important to emphasize the size, we shall write In for n× n identitymatrix, such as

[1 00 1

], I3 =

1 0 00 1 00 0 1

, I4 =

1 0 0 00 1 0 00 0 1 00 0 0 1

, and so on.

2.2 Inverse Matrix

Definition (2.2.1)

An identity matrix , denoted by I, is a square matrix with 1’s on the maindoagonal and 0’s off the main diagonal.

If it is important to emphasize the size, we shall write In for n× n identitymatrix, such as

[1 00 1

], I3 =

1 0 00 1 00 0 1

, I4 =

1 0 0 00 1 0 00 0 1 00 0 0 1

, and so on.

Example (2.2.1)

Let A = [aij ] be 2× 3 matrix. Find I2A and AI3.

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

Example (2.2.1)

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

Example (2.2.1)

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

Example (2.2.1)

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

Example (2.2.1)

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

Example (2.2.1)

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

Example (2.2.1)

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

Example (2.2.1)

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

Example (2.2.1)

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

Example (2.2.1)

Solution

[a11 a12 a13a21 a22 a23

[1 00 1

] [a11 a12 a13a21 a22 a23

[a11 + 0 a12 + 0 a13 + 00 + a21 0 + a22 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

[a11 a12 a13a21 a22 a23

] 1 0 00 1 00 0 1

[a11 + 0 + 0 0 + a12 + 0 0 + 0 + a13a21 + 0 + 0 0 + a22 + 0 0 + 0 + a23

[a11 a12 a13a21 a22 a23

]= A #

Invertible Matrix

Definition (2.2.2)

Let A be a square matrix. If a matrix B of the same size to A can be foundsuch that

AB = I = BA,

then A is said to be invertible and B is called an inverse of A. If no such

matrix B can be found, then A is said to be singular .

Example (2.2.2)

Show that B is an inverse of A.

[3 51 2

]and B =

[2 −5−1 3

Proof .

[3 51 2

] [2 −5−1 3

[6− 5 15− 152− 2 −5 + 6

[1 00 1

[2 −5−1 3

] [3 51 2

[6− 5 10− 10−3 + 3 −5 + 6

[1 00 1

∴ AB = I = BA. Thus, B is an inverse of A.

Example (2.2.2)

[3 51 2

]and B =

[2 −5−1 3

Proof .

[3 51 2

] [2 −5−1 3

[6− 5 15− 152− 2 −5 + 6

[1 00 1

[2 −5−1 3

] [3 51 2

[6− 5 10− 10−3 + 3 −5 + 6

[1 00 1

Example (2.2.2)

[3 51 2

]and B =

[2 −5−1 3

Proof .

[3 51 2

] [2 −5−1 3

[6− 5 15− 152− 2 −5 + 6

[1 00 1

[2 −5−1 3

] [3 51 2

[6− 5 10− 10−3 + 3 −5 + 6

[1 00 1

Example (2.2.2)

[3 51 2

]and B =

[2 −5−1 3

Proof .

[3 51 2

] [2 −5−1 3

[6− 5 15− 152− 2 −5 + 6

[1 00 1

[2 −5−1 3

] [3 51 2

[6− 5 10− 10−3 + 3 −5 + 6

[1 00 1

Example (2.2.2)

[3 51 2

]and B =

[2 −5−1 3

Proof .

[3 51 2

] [2 −5−1 3

[6− 5 15− 152− 2 −5 + 6

[1 00 1

[2 −5−1 3

] [3 51 2

[6− 5 10− 10−3 + 3 −5 + 6

[1 00 1

Example (2.2.2)

[3 51 2

]and B =

[2 −5−1 3

Proof .

[3 51 2

] [2 −5−1 3

[6− 5 15− 152− 2 −5 + 6

[1 00 1

[2 −5−1 3

] [3 51 2

[6− 5 10− 10−3 + 3 −5 + 6

[1 00 1

Example (2.2.2)

[3 51 2

]and B =

[2 −5−1 3

Proof .

[3 51 2

] [2 −5−1 3

[6− 5 15− 152− 2 −5 + 6

[1 00 1

[2 −5−1 3

] [3 51 2

[6− 5 10− 10−3 + 3 −5 + 6

[1 00 1

Example (2.2.2)

[3 51 2

]and B =

[2 −5−1 3

Proof .

[3 51 2

] [2 −5−1 3

[6− 5 15− 152− 2 −5 + 6

[1 00 1

[2 −5−1 3

] [3 51 2

[6− 5 10− 10−3 + 3 −5 + 6

[1 00 1

Uniqueness of Inverse

Theorem (2.2.1)

If a matrix is invertible, then it has a unique inverse.

Proof .

Let B1 and B2 be inverses of A. Then AB1 = I = B1A and AB2 = I = B2A. We obtain

B2(AB1) = B2(I)

(B2A)B1 = B2

(I)B1 = B2

B1 = B2

We can say of the inverse of an invertible matrix A denoted by A−1 ; that is

A−1A = I = AA−1

Theorem (2.2.1)

Proof .

B2(AB1) = B2(I)

(B2A)B1 = B2

(I)B1 = B2

B1 = B2

A−1A = I = AA−1

Theorem (2.2.1)

Proof .

B2(AB1) = B2(I)

(B2A)B1 = B2

(I)B1 = B2

B1 = B2

A−1A = I = AA−1

Inverse of 2× 2 Matrix

Theorem (2.2.2)

The matrix

[a bc d

]is invertible if ad− bc ̸= 0, in which case the inverse is given by the formula

A−1 =1

ad− bc

[d −b−c a

Proof .

AA−1 =

[a bc d

ad− bc

[d −b−c a

ad− bc

[ad− bc −ab+ abcd− cd −cb+ ad

[1 00 1

]A−1A =

ad− bc

[d −b−c a

] [a bc d

ad− bc

[ad− bc bd− bd−ac+ ac −bc+ ad

[1 00 1

]∴ AA−1 = I = A−1A. Thus, A−1 is an inverse of A.

Theorem (2.2.2)

The matrix

[a bc d

A−1 =1

ad− bc

[d −b−c a

]Proof .

AA−1 =

[a bc d

ad− bc

[d −b−c a

ad− bc

[1 00 1

]A−1A =

ad− bc

[d −b−c a

] [a bc d

ad− bc

[1 00 1

∴ AA−1 = I = A−1A. Thus, A−1 is an inverse of A.

Theorem (2.2.2)

The matrix

[a bc d

A−1 =1

ad− bc

[d −b−c a

]Proof .

AA−1 =

[a bc d

ad− bc

[d −b−c a

ad− bc

[1 00 1

]A−1A =

ad− bc

[d −b−c a

] [a bc d

ad− bc

[1 00 1

]∴ AA−1 = I = A−1A. Thus, A−1 is an inverse of A.

Example (2.2.3)

Find A−1, B−1, A−1B−1, B−1A−1 and (AB)−1.

[1 42 7

]and B =

[−3 −51 2

Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

(−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

[4 −3−1 1

Example (2.2.3)

[1 42 7

]and B =

[−3 −51 2

]Solution

A−1 = 17(1)−4(2)

[7 −4−2 1

]= (−1)

[7 −4−2 1

[−7 42 −1

B−1 = 12(−3)−1(−5)

[2 −(−5)−1 −3

]= (−1)

[2 5−1 −3

[−2 −51 3

A−1B−1=

[−7 42 −1

] [−2 −51 3

[14 + 4 35 + 12−4− 1 −10− 3

[18 47−5 −13

B−1A−1=

[−2 −51 3

] [−7 42 −1

[14− 10 −8 + 5−7 + 6 4− 3

[4 −3−1 1

[1 42 7

] [−3 −51 2

[−3 + 4 −5 + 8−6 + 7 −10 + 14

[1 31 4

(AB)−1 = 11(4)−(−1)(−3)

[1 −3−1 4

]= (1)

[4 −3−1 1

Power of Matrix

Theorem (2.2.3)

Let A and B be square matrices of the same size. If A and B are invertible,then

(AB)−1 = B−1A−1

Proof.

Let A and B be square matrices of the same size. Assume that A and B are invertible. Then

(B−1A−1)(AB) = B−1(A−1A)B = B−1(I)B

= B−1B = I

(AB)(B−1A−1) = A(BB−1)A−1 = A(I)A−1

= AA−1 = I

Thus, B−1A−1 is the inverse of AB. Hence, (AB)−1 = B−1A−1.

Power of Matrix

Theorem (2.2.3)

(AB)−1 = B−1A−1

Proof.

Let A and B be square matrices of the same size. Assume that A and B are invertible.

(B−1A−1)(AB) = B−1(A−1A)B = B−1(I)B

= B−1B = I

(AB)(B−1A−1) = A(BB−1)A−1 = A(I)A−1

= AA−1 = I

Power of Matrix

Theorem (2.2.3)

(AB)−1 = B−1A−1

Proof.

(B−1A−1)(AB) = B−1(A−1A)B = B−1(I)B

= B−1B = I

(AB)(B−1A−1) = A(BB−1)A−1 = A(I)A−1

= AA−1 = I

Power of Matrix

Theorem (2.2.3)

(AB)−1 = B−1A−1

Proof.

(B−1A−1)(AB) = B−1(A−1A)B = B−1(I)B

= B−1B = I

(AB)(B−1A−1) = A(BB−1)A−1 = A(I)A−1

= AA−1 = I

Power of Matrix

Definition (2.2.3)

Let A be a square matrix. Then we define the nonnegative integer n power of A to be

the nonnegative integer power of A to be

A0 = I and An = AA · · ·A︸︷︷︸n−factors

for n > 0

if A is invertibleA−n = (A−1)n = A−1A−1 · · ·A−1︸︷︷︸

n−factors

Theorem (2.2.4)

Let A be invertible matrix and n and m be integers. Then

1 AmAn = Am+n

2 (An)m = (Am)n = Anm

Theorem (2.2.5)

Let A be invertible matrix and n be an integer. Then

1 (A−1)−1 = A

2 (An)−1 = (A−1)n

3 (AT )−1 = (A−1)T

4 (kA)−1 = 1kA−1 where k ∈ R and k ̸= 0

Theorem (2.2.4)

Let A be invertible matrix and n and m be integers. Then

1 AmAn = Am+n

2 (An)m = (Am)n = Anm

Theorem (2.2.5)

Let A be invertible matrix and n be an integer. Then

1 (A−1)−1 = A

2 (An)−1 = (A−1)n

3 (AT )−1 = (A−1)T

4 (kA)−1 = 1kA−1 where k ∈ R and k ̸= 0

Example (2.2.4)

Find A2, (AT )−1, A−3 and (2A)−4 if A =

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1

= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Example (2.2.4)

[1 21 3

Solution

A2 = AA =

[1 21 3

] [1 21 3

[1 + 2 2 + 61 + 3 2 + 9

[3 84 11

(AT )−1 =

([1 21 3

]T)−1

[1 12 3

= 11(3)−1(2)

[3 −2−1 1

A−1 = 11(3)−1(2)

[3 −2−1 1

]A−3 = A−1A−1A−1= A−1

[3 −2−1 1

] [3 −2−1 1

]= A−1

[9 + 2 −6− 2−3− 1 2 + 1

[3 −2−1 1

] [11 −8−4 3

[33 + 8 −24− 6−11− 4 8 + 3

[41 −30−15 11

(2A)−4 = 2−4A−4 = 116

A−1A−3 = 116

[3 −2−1 1

] [41 −30−15 11

[123 + 30 −90− 22−41− 15 30 + 11

[153 −112−56 41

[ 15316

− 72

Polynomial Matrix

Definition (2.2.4)

Let A be a square matrix. If

p(x) = a0 + a1x+ a2x2 + · · ·+ anx

is any polynomial, the we define

p(A) = a0I + a1A+ a2A2 + · · ·+ anA

Polynomial Matrix

Definition (2.2.4)

Let A be a square matrix. If

p(x) = a0 + a1x+ a2x2 + · · ·+ anx

is any polynomial, the we define

p(A) = a0I + a1A+ a2A2 + · · ·+ anA

Example (2.2.5)

Let p(x) = 2x2 − 3x+ 4 and A =

[−1 20 3

]. Find p(A).

Solution

p(A) = 2A2 − 3A+ 4I

[−1 20 3

]2− 3

[−1 20 3

[1 00 1

[−1 20 3

] [−1 20 3

[3 −60 −9

[4 00 4

[1 + 0 −2 + 60 + 0 0 + 9

[7 −60 −5

[1 40 9

[7 −60 −5

[2 80 18

[7 −60 −5

[9 20 13

Take a BreakFor 10 Minutes

Elementary Matrix

Definition (2.2.5)

An n× n matrix is called an elementary matrix if it can be obtained from then× n identity matrix In by performing a single elementary row operation.

Example (2.2.6)

Listed below are elementary matrices. What is a single ERO produce them ?

[1 00 2

[1 00 1

]2R2−−−→

[1 00 2

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

R3−5R2−−−−−−−→

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

(1)R1−−−−−→

1 0 00 1 00 0 1

0 0 0 10 1 0 00 0 1 01 0 0 0

1 0 0 00 1 0 00 0 1 00 0 0 1

R14−−−→

0 0 0 10 1 0 00 0 1 01 0 0 0

Elementary Matrix

Definition (2.2.5)

Example (2.2.6)

[1 00 2

[1 00 1

]2R2−−−→

[1 00 2

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

R3−5R2−−−−−−−→

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

(1)R1−−−−−→

1 0 00 1 00 0 1

0 0 0 10 1 0 00 0 1 01 0 0 0

1 0 0 00 1 0 00 0 1 00 0 0 1

R14−−−→

0 0 0 10 1 0 00 0 1 01 0 0 0

Elementary Matrix

Definition (2.2.5)

Example (2.2.6)

[1 00 2

[1 00 1

]2R2−−−→

[1 00 2

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

R3−5R2−−−−−−−→

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

(1)R1−−−−−→

1 0 00 1 00 0 1

0 0 0 10 1 0 00 0 1 01 0 0 0

1 0 0 00 1 0 00 0 1 00 0 0 1

R14−−−→

0 0 0 10 1 0 00 0 1 01 0 0 0

Elementary Matrix

Definition (2.2.5)

Example (2.2.6)

[1 00 2

[1 00 1

]2R2−−−→

[1 00 2

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

R3−5R2−−−−−−−→

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

(1)R1−−−−−→

1 0 00 1 00 0 1

0 0 0 10 1 0 00 0 1 01 0 0 0

1 0 0 00 1 0 00 0 1 00 0 0 1

R14−−−→

0 0 0 10 1 0 00 0 1 01 0 0 0

Elementary Matrix

Definition (2.2.5)

Example (2.2.6)

[1 00 2

[1 00 1

]2R2−−−→

[1 00 2

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

R3−5R2−−−−−−−→

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

(1)R1−−−−−→

1 0 00 1 00 0 1

0 0 0 10 1 0 00 0 1 01 0 0 0

1 0 0 00 1 0 00 0 1 00 0 0 1

R14−−−→

0 0 0 10 1 0 00 0 1 01 0 0 0

Elementary Matrix

Definition (2.2.5)

Example (2.2.6)

[1 00 2

[1 00 1

]2R2−−−→

[1 00 2

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

R3−5R2−−−−−−−→

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

(1)R1−−−−−→

1 0 00 1 00 0 1

0 0 0 10 1 0 00 0 1 01 0 0 0

1 0 0 00 1 0 00 0 1 00 0 0 1

R14−−−→

0 0 0 10 1 0 00 0 1 01 0 0 0

Elementary Matrix

Definition (2.2.5)

Example (2.2.6)

[1 00 2

[1 00 1

]2R2−−−→

[1 00 2

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

R3−5R2−−−−−−−→

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

(1)R1−−−−−→

1 0 00 1 00 0 1

0 0 0 10 1 0 00 0 1 01 0 0 0

1 0 0 00 1 0 00 0 1 00 0 0 1

R14−−−→

0 0 0 10 1 0 00 0 1 01 0 0 0

Elementary Matrix

Definition (2.2.5)

Example (2.2.6)

[1 00 2

[1 00 1

]2R2−−−→

[1 00 2

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

R3−5R2−−−−−−−→

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

(1)R1−−−−−→

1 0 00 1 00 0 1

0 0 0 10 1 0 00 0 1 01 0 0 0

1 0 0 00 1 0 00 0 1 00 0 0 1

R14−−−→

0 0 0 10 1 0 00 0 1 01 0 0 0

Elementary Matrix

Definition (2.2.5)

Example (2.2.6)

[1 00 2

[1 00 1

]2R2−−−→

[1 00 2

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

R3−5R2−−−−−−−→

1 0 00 1 00 −5 1

1 0 00 1 00 0 1

(1)R1−−−−−→

1 0 00 1 00 0 1

0 0 0 10 1 0 00 0 1 01 0 0 0

1 0 0 00 1 0 00 0 1 00 0 0 1

R14−−−→

0 0 0 10 1 0 00 0 1 01 0 0 0

Example (2.2.7)

For matrices E =

1 0 00 1 02 0 1

and A =

1 1 2 32 −1 3 51 3 7 0

Compute EA.

Solution

1 0 00 1 02 0 1

1 1 2 32 −1 3 51 3 7 0

1 1 2 32 −1 3 5

2(1) + 1 2(1) + 3 2(2) + 7 2(3) + 0

1 1 2 32 −1 3 53 5 11 6

1 0 00 1 00 0 1

R3+2R1−−−−−−→

1 0 00 1 02 0 1

1 1 2 32 −1 3 51 3 7 0

R3+2R1−−−−−−→

1 1 2 32 −1 3 53 5 11 6

Example (2.2.7)

For matrices E =

1 0 00 1 02 0 1

and A =

1 1 2 32 −1 3 51 3 7 0

Compute EA.

Solution

1 0 00 1 02 0 1

1 1 2 32 −1 3 51 3 7 0

1 1 2 32 −1 3 5

2(1) + 1 2(1) + 3 2(2) + 7 2(3) + 0

1 1 2 32 −1 3 53 5 11 6

1 0 00 1 00 0 1

R3+2R1−−−−−−→

1 0 00 1 02 0 1

1 1 2 32 −1 3 51 3 7 0

R3+2R1−−−−−−→

1 1 2 32 −1 3 53 5 11 6

Example (2.2.7)

For matrices E =

1 0 00 1 02 0 1

and A =

1 1 2 32 −1 3 51 3 7 0

Compute EA.

Solution

1 0 00 1 02 0 1

1 1 2 32 −1 3 51 3 7 0

1 1 2 32 −1 3 5

2(1) + 1 2(1) + 3 2(2) + 7 2(3) + 0

1 1 2 32 −1 3 53 5 11 6

1 0 00 1 00 0 1

R3+2R1−−−−−−→

1 0 00 1 02 0 1

1 1 2 32 −1 3 51 3 7 0

R3+2R1−−−−−−→

1 1 2 32 −1 3 53 5 11 6

Example (2.2.7)

For matrices E =

1 0 00 1 02 0 1

and A =

1 1 2 32 −1 3 51 3 7 0

Compute EA.

Solution

1 0 00 1 02 0 1

1 1 2 32 −1 3 51 3 7 0

1 1 2 32 −1 3 5

2(1) + 1 2(1) + 3 2(2) + 7 2(3) + 0

1 1 2 32 −1 3 53 5 11 6

1 0 00 1 00 0 1

R3+2R1−−−−−−→

1 0 00 1 02 0 1

1 1 2 32 −1 3 51 3 7 0

R3+2R1−−−−−−→

1 1 2 32 −1 3 53 5 11 6

Theorem (2.2.6)

If the elementary matrix E results from performing a certain row operation on Imand if A is an m× n matrix, then the product EA is the matrix that results whenthis same row operation is performing on A.

If an ERO is applied to an identity matrix to produce an elementary matrix E, thenthere is a second row operation that, when applied to E, produces I back again.

Row operation on I that produce E Row operation on E that reproduce I

Multiply row i by c ̸= 0 Multiply row i by 1c

Interchang row i and j Interchang row i and j

Add c times row i to row j Add −c times row i to row j

[1 00 1

]5R2−−→

[1 00 5

]15R2−−−→

[1 00 1

][1 00 1

]R12−−→

[0 11 0

]R12−−→

[1 00 1

][1 00 1

]R1+7R2−−−−−→

[1 70 1

]R1−7R2−−−−−→

[1 00 1

Theorem (2.2.6)

[1 00 1

]5R2−−→

[1 00 5

]15R2−−−→

[1 00 1

][1 00 1

]R12−−→

[0 11 0

]R12−−→

[1 00 1

][1 00 1

]R1+7R2−−−−−→

[1 70 1

]R1−7R2−−−−−→

[1 00 1

Theorem (2.2.6)

[1 00 1

]5R2−−→

[1 00 5

]15R2−−−→

[1 00 1

]R12−−→

[0 11 0

]R12−−→

[1 00 1

][1 00 1

]R1+7R2−−−−−→

[1 70 1

]R1−7R2−−−−−→

[1 00 1

Theorem (2.2.6)

[1 00 1

]5R2−−→

[1 00 5

]15R2−−−→

[1 00 1

][1 00 1

]R12−−→

[0 11 0

]R12−−→

[1 00 1

]R1+7R2−−−−−→

[1 70 1

]R1−7R2−−−−−→

[1 00 1

Theorem (2.2.6)

[1 00 1

]5R2−−→

[1 00 5

]15R2−−−→

[1 00 1

][1 00 1

]R12−−→

[0 11 0

]R12−−→

[1 00 1

][1 00 1

]R1+7R2−−−−−→

[1 70 1

]R1−7R2−−−−−→

[1 00 1

]Thanatyod Jampawai, Ph.D. MAT2305 LINEAR ALGEBRA :Week2 37 / 60

Theorem (2.2.7)

Every elementary matrix is invertible, and the inverse is also an elementary matrix.

Theorem (2.2.8)

Let A be an n× n matrix. Then the following statements are equivalent.

1 A is invertible.

2 The RREF of A from A is In.

3 A is expressible as a product of elementary matrices.

We estabish a method for determining the inverse of an n× n invertible matrix A. We can findelementary matrices E1, E2, .., Ek such that

Ek · · ·E2E1A = In.

Multiplying this equation on the right by A−1 yields

A−1 = Ek · · ·E2E1In

Theorem (2.2.7)

Theorem (2.2.8)

1 A is invertible.

Ek · · ·E2E1A = In.

A−1 = Ek · · ·E2E1In

Theorem (2.2.7)

Theorem (2.2.8)

1 A is invertible.

Ek · · ·E2E1A = In.

A−1 = Ek · · ·E2E1In

Theorem (2.2.7)

Theorem (2.2.8)

1 A is invertible.

Ek · · ·E2E1A = In.

A−1 = Ek · · ·E2E1In

Theorem (2.2.7)

Theorem (2.2.8)

1 A is invertible.

Ek · · ·E2E1A = In.

A−1 = Ek · · ·E2E1In

Example (2.2.8)

Find inverse of A =

1 2 32 5 31 0 8

Solution

[A|I] =

1 2 3 1 0 02 5 3 0 1 01 0 8 0 0 1

R2−2R1−−−−−−→R3−R1

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

−→

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

R1−2R2−−−−−−→R3+2R2

1 0 9 5 −2 00 1 −3 −2 1 00 0 −1 −5 2 1

Example (2.2.8)

Find inverse of A =

1 2 32 5 31 0 8

Solution

[A|I] =

1 2 3 1 0 02 5 3 0 1 01 0 8 0 0 1

R2−2R1−−−−−−→R3−R1

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

−→

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

R1−2R2−−−−−−→R3+2R2

1 0 9 5 −2 00 1 −3 −2 1 00 0 −1 −5 2 1

Example (2.2.8)

Find inverse of A =

1 2 32 5 31 0 8

Solution

[A|I] =

1 2 3 1 0 02 5 3 0 1 01 0 8 0 0 1

R2−2R1−−−−−−→R3−R1

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

−→

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

R1−2R2−−−−−−→R3+2R2

1 0 9 5 −2 00 1 −3 −2 1 00 0 −1 −5 2 1

Example (2.2.8)

Find inverse of A =

1 2 32 5 31 0 8

Solution

[A|I] =

1 2 3 1 0 02 5 3 0 1 01 0 8 0 0 1

R2−2R1−−−−−−→R3−R1

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

−→

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

R1−2R2−−−−−−→R3+2R2

1 0 9 5 −2 00 1 −3 −2 1 00 0 −1 −5 2 1

Example (2.2.8)

Find inverse of A =

1 2 32 5 31 0 8

Solution

[A|I] =

1 2 3 1 0 02 5 3 0 1 01 0 8 0 0 1

R2−2R1−−−−−−→R3−R1

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

−→

1 2 3 1 0 00 1 −3 −2 1 00 −2 5 −1 0 1

R1−2R2−−−−−−→R3+2R2

1 0 9 5 −2 00 1 −3 −2 1 00 0 −1 −5 2 1

Solution

−→

1 0 9 5 −2 00 1 −3 −2 1 00 0 −1 −5 2 1

(−1)R3−−−−−→

1 0 9 5 −2 00 1 −3 −2 1 00 0 1 5 −2 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −40 16 90 1 0 13 −5 −30 0 1 5 −2 −1

A−1 =

−40 16 913 −5 −35 −2 −1

Solution

−→

1 0 9 5 −2 00 1 −3 −2 1 00 0 −1 −5 2 1

(−1)R3−−−−−→

1 0 9 5 −2 00 1 −3 −2 1 00 0 1 5 −2 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −40 16 90 1 0 13 −5 −30 0 1 5 −2 −1

A−1 =

−40 16 913 −5 −35 −2 −1

Solution

−→

1 0 9 5 −2 00 1 −3 −2 1 00 0 −1 −5 2 1

(−1)R3−−−−−→

1 0 9 5 −2 00 1 −3 −2 1 00 0 1 5 −2 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −40 16 90 1 0 13 −5 −30 0 1 5 −2 −1

A−1 =

−40 16 913 −5 −35 −2 −1

Solution

−→

1 0 9 5 −2 00 1 −3 −2 1 00 0 −1 −5 2 1

(−1)R3−−−−−→

1 0 9 5 −2 00 1 −3 −2 1 00 0 1 5 −2 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −40 16 90 1 0 13 −5 −30 0 1 5 −2 −1

A−1 =

−40 16 913 −5 −35 −2 −1

Recheck

AA−1 =

1 2 32 5 31 0 8

−40 16 913 −5 −35 −2 −1

−40 + 26 + 15 16− 10− 6 9− 6− 3−80 + 65 + 15 32− 25− 6 18− 15− 3−40 + 0 + 40 16 + 0− 16 9 + 0− 8

1 0 00 1 00 0 1

A−1A =

−40 16 913 −5 −35 −2 −1

1 2 32 5 31 0 8

−40 + 32 + 9 −80 + 80 + 0 −120 + 48 + 7213− 10− 3 26− 25 + 0 39− 15− 245− 4− 1 10− 10 + 0 15− 6− 8

1 0 00 1 00 0 1

Example (2.2.9)

Find inverse of A =

1 2 3 00 1 −4 52 0 2 00 4 0 1

Homework

A−1 =1

−38 −4 −49 20−8 −4 4 2038 4 −19 −2032 16 −16 −20

Example (2.2.10)

Find inverse of A =

1 0 0 01 1 0 01 2 1 01 3 2 1

Solution

1 0 0 0 1 0 0 01 1 0 0 0 1 0 01 2 1 0 0 0 1 01 3 2 1 0 0 0 1

R3−R1,R2−R1−−−−−−−−−−−−→R4−R1

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 2 1 0 −1 0 1 00 3 2 1 −1 0 0 1

R3−2R1−−−−−−−→R4−3R2

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 2 1 2 −3 0 1

R4−2R3−−−−−−−→

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 0 1 0 1 −2 1

Example (2.2.10)

Find inverse of A =

1 0 0 01 1 0 01 2 1 01 3 2 1

Solution

1 0 0 0 1 0 0 01 1 0 0 0 1 0 01 2 1 0 0 0 1 01 3 2 1 0 0 0 1

R3−R1,R2−R1−−−−−−−−−−−−→R4−R1

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 2 1 0 −1 0 1 00 3 2 1 −1 0 0 1

R3−2R1−−−−−−−→R4−3R2

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 2 1 2 −3 0 1

R4−2R3−−−−−−−→

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 0 1 0 1 −2 1

Example (2.2.10)

Find inverse of A =

1 0 0 01 1 0 01 2 1 01 3 2 1

Solution

1 0 0 0 1 0 0 01 1 0 0 0 1 0 01 2 1 0 0 0 1 01 3 2 1 0 0 0 1

R3−R1,R2−R1−−−−−−−−−−−−→R4−R1

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 2 1 0 −1 0 1 00 3 2 1 −1 0 0 1

R3−2R1−−−−−−−→R4−3R2

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 2 1 2 −3 0 1

R4−2R3−−−−−−−→

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 0 1 0 1 −2 1

Example (2.2.10)

Find inverse of A =

1 0 0 01 1 0 01 2 1 01 3 2 1

Solution

1 0 0 0 1 0 0 01 1 0 0 0 1 0 01 2 1 0 0 0 1 01 3 2 1 0 0 0 1

R3−R1,R2−R1−−−−−−−−−−−−→R4−R1

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 2 1 0 −1 0 1 00 3 2 1 −1 0 0 1

R3−2R1−−−−−−−→R4−3R2

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 2 1 2 −3 0 1

R4−2R3−−−−−−−→

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 0 1 0 1 −2 1

Example (2.2.10)

Find inverse of A =

1 0 0 01 1 0 01 2 1 01 3 2 1

Solution

1 0 0 0 1 0 0 01 1 0 0 0 1 0 01 2 1 0 0 0 1 01 3 2 1 0 0 0 1

R3−R1,R2−R1−−−−−−−−−−−−→R4−R1

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 2 1 0 −1 0 1 00 3 2 1 −1 0 0 1

R3−2R1−−−−−−−→R4−3R2

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 2 1 2 −3 0 1

R4−2R3−−−−−−−→

1 0 0 0 1 0 0 00 1 0 0 −1 1 0 00 0 1 0 1 −2 1 00 0 0 1 0 1 −2 1

Example (2.2.11)

Find inverse of A =

1 2 −3 00 1 5 06 0 1 00 0 0 1

Homework

A−1 =1

1 −2 13 030 19 −5 0−6 12 1 00 0 0 79

Example (2.2.12)

Show that matrix A is not invertible (singular) A =

1 6 42 4 −1−1 2 5

Solution 1 6 4 1 0 02 4 −1 0 1 0−1 2 5 0 0 1

R2−2R1−−−−−−→R3+R1

1 6 4 1 0 00 −8 −9 −2 1 00 8 9 1 0 1

R3+R2−−−−−→

1 6 4 1 0 00 −8 −9 −2 1 00 0 0 −1 1 1

Example (2.2.12)

1 6 42 4 −1−1 2 5

Solution 1 6 4 1 0 0

2 4 −1 0 1 0−1 2 5 0 0 1

R2−2R1−−−−−−→R3+R1

1 6 4 1 0 00 −8 −9 −2 1 00 8 9 1 0 1

R3+R2−−−−−→

1 6 4 1 0 00 −8 −9 −2 1 00 0 0 −1 1 1

Example (2.2.12)

1 6 42 4 −1−1 2 5

2 4 −1 0 1 0−1 2 5 0 0 1

R2−2R1−−−−−−→R3+R1

1 6 4 1 0 00 −8 −9 −2 1 00 8 9 1 0 1

R3+R2−−−−−→

1 6 4 1 0 00 −8 −9 −2 1 00 0 0 −1 1 1

Example (2.2.12)

1 6 42 4 −1−1 2 5

2 4 −1 0 1 0−1 2 5 0 0 1

R2−2R1−−−−−−→R3+R1

1 6 4 1 0 00 −8 −9 −2 1 00 8 9 1 0 1

R3+R2−−−−−→

1 6 4 1 0 00 −8 −9 −2 1 00 0 0 −1 1 1

Matrices Linear Systems with Invertibility

2.3 Linear Systems with Invertibility

Marix multiplication has an important application to system of linearequations. Consider any linear system of m linear equation in n unknowns.

a11x1 + a12x2 + ...+ a12xn = b1

a21x1 + a22x2 + ...+ a2nxn = b2

...am1x1 + am2x2 + ...+ amnxn = bm

We can replace the m equations in this system by product matrices.a11 a12 ... a12a21 a22 ... a2n

......

...am1 am2 ... amn

b1b2...bm

If we designate matrices by A, x and b, respectively, the original systembecomes to

Ax = b

a11x1 + a12x2 + ...+ a12xn = b1

a21x1 + a22x2 + ...+ a2nxn = b2

...am1x1 + am2x2 + ...+ amnxn = bm

......

...am1 am2 ... amn

b1b2...bm

Ax = b

a11x1 + a12x2 + ...+ a12xn = b1

a21x1 + a22x2 + ...+ a2nxn = b2

...am1x1 + am2x2 + ...+ amnxn = bm

......

...am1 am2 ... amn

b1b2...bm

Ax = bThanatyod Jampawai, Ph.D. MAT2305 LINEAR ALGEBRA :Week2 46 / 60

The matrix A is called the cofficient matrix of the system. The argumentedmatrix for the system is obatined by

[A | b ] =

a11 a12 · · · a1n b1a21 a22 · · · a2n b2

......

am1 am2 · · · amn bm

Theorem (2.3.1)

Every system of linear equations has either no solutions, exactly one solution,or infinitely many solutions.

The matrix A is called the cofficient matrix of the system. The argumentedmatrix for the system is obatined by

[A | b ] =

a11 a12 · · · a1n b1a21 a22 · · · a2n b2

......

am1 am2 · · · amn bm

Theorem (2.3.1)

Every system of linear equations has either no solutions, exactly one solution,or infinitely many solutions.

Theorem (2.3.2)

Let A be an n× n matrix and let b be an n× 1 matrix. If the linear equationsAx = b has exactly one solution, then, namely,

x = A−1b

Example (2.3.1)

Use theorem 2.3.2 to find the system of linear equations

{3x1 + 2x2 = 5

7x1 + 5x2 = 3and (b)

x1 + 2x2 + 3x3 = 5

2x1 + 5x2 + 3x3 = 3

x1 + 8x3 = 11

Theorem (2.3.2)

Let A be an n× n matrix and let b be an n× 1 matrix. If the linear equationsAx = b has exactly one solution, then, namely,

x = A−1b

Example (2.3.1)

Use theorem 2.3.2 to find the system of linear equations

{3x1 + 2x2 = 5

7x1 + 5x2 = 3and (b)

x1 + 2x2 + 3x3 = 5

2x1 + 5x2 + 3x3 = 3

x1 + 8x3 = 11

Solution

{3x1 + 2x2 = 5

7x1 + 5x2 = 3↔

[3 27 5

]−1 [53

[5 −2−7 3

[25− 6−35 + 9

[19−26

]Hence, x1 = 19, x2 = −26 #

Solution

{3x1 + 2x2 = 5

7x1 + 5x2 = 3↔

[3 27 5

]Then,

[3 27 5

]−1 [53

[5 −2−7 3

[25− 6−35 + 9

[19−26

Hence, x1 = 19, x2 = −26 #

Solution

{3x1 + 2x2 = 5

7x1 + 5x2 = 3↔

[3 27 5

]Then,

[3 27 5

]−1 [53

[5 −2−7 3

[25− 6−35 + 9

[19−26

]Hence, x1 = 19, x2 = −26 #

Solution

x1 + 2x2 + 3x3 = 5

2x1 + 5x2 + 3x3 = 3

x1 + 8x3 = 11

1 2 32 5 31 0 8

−1 5311

−40 16 913 −5 −35 −2 −1

−200 + 48 + 9965− 15− 3325− 6− 11

−53178

Hence, x1 = −53, x2 = 17, x3 = 8 #

Solution

x1 + 2x2 + 3x3 = 5

2x1 + 5x2 + 3x3 = 3

x1 + 8x3 = 11

1 2 32 5 31 0 8

−1 5311

−40 16 913 −5 −35 −2 −1

−200 + 48 + 9965− 15− 3325− 6− 11

−53178

Hence, x1 = −53, x2 = 17, x3 = 8 #

Solution

x1 + 2x2 + 3x3 = 5

2x1 + 5x2 + 3x3 = 3

x1 + 8x3 = 11

1 2 32 5 31 0 8

−1 5311

−40 16 913 −5 −35 −2 −1

−200 + 48 + 9965− 15− 3325− 6− 11

−53178

Hence, x1 = −53, x2 = 17, x3 = 8 #

Linear systems with a common coefficient matrix: One is concerned withsolving a seguence of systems

Ax = b1, Ax = b2, Ax = b3, ..., Ax = bk

If A is invertible, then the solutions

x1 = A−1b1, x2 = A−1b2, x3 = A−1b3, ..., xk = A−1bk

can be obtained with one matrix inversion and k matrix multiplications. Amore effient method is to form the matrix (argumented matrix)

[ A | b1 | b2 | · · · | bk ]

Ax = b1, Ax = b2, Ax = b3, ..., Ax = bk

x1 = A−1b1, x2 = A−1b2, x3 = A−1b3, ..., xk = A−1bk

[ A | b1 | b2 | · · · | bk ]

Ax = b1, Ax = b2, Ax = b3, ..., Ax = bk

x1 = A−1b1, x2 = A−1b2, x3 = A−1b3, ..., xk = A−1bk

[ A | b1 | b2 | · · · | bk ]

Example (2.3.2)

Use Gauss-Jodan elimination to find two systems of linear equations

x1 + 2x2 + 3x3 = 4

2x1 + 5x2 + 3x3 = 5

x1 + 8x3 = 6

x1 + 2x2 + 3x3 = 1

2x1 + 5x2 + 3x3 = 6

x1 + 8x3 = −6

1 2 3 4 12 5 3 5 61 0 8 6 −6

Example (2.3.2)

Use Gauss-Jodan elimination to find two systems of linear equations

x1 + 2x2 + 3x3 = 4

2x1 + 5x2 + 3x3 = 5

x1 + 8x3 = 6

x1 + 2x2 + 3x3 = 1

2x1 + 5x2 + 3x3 = 6

x1 + 8x3 = −6

1 2 3 4 12 5 3 5 61 0 8 6 −6

Solution 1 2 3 4 12 5 3 5 61 0 8 6 −6

R2−2R1−−−−−−→R3−R1

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

−→

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

R1−2R2−−−−−−→R3+2R2

1 0 9 10 −70 1 −3 −3 40 0 −1 −4 1

(−1)R3−−−−−→

1 0 9 10 −70 1 −3 −3 40 0 1 4 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −26 20 1 0 9 10 0 1 4 −1

Thus, x1 = −26, x2 = 9, x3 = 4 is the solution of (a) and x1 = 2, x2 = 1, x3 = −1 is thesolution of (b). #

Solution 1 2 3 4 12 5 3 5 61 0 8 6 −6

R2−2R1−−−−−−→R3−R1

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

−→

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

R1−2R2−−−−−−→R3+2R2

1 0 9 10 −70 1 −3 −3 40 0 −1 −4 1

(−1)R3−−−−−→

1 0 9 10 −70 1 −3 −3 40 0 1 4 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −26 20 1 0 9 10 0 1 4 −1

Solution 1 2 3 4 12 5 3 5 61 0 8 6 −6

R2−2R1−−−−−−→R3−R1

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

−→

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

R1−2R2−−−−−−→R3+2R2

1 0 9 10 −70 1 −3 −3 40 0 −1 −4 1

(−1)R3−−−−−→

1 0 9 10 −70 1 −3 −3 40 0 1 4 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −26 20 1 0 9 10 0 1 4 −1

Solution 1 2 3 4 12 5 3 5 61 0 8 6 −6

R2−2R1−−−−−−→R3−R1

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

−→

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

R1−2R2−−−−−−→R3+2R2

1 0 9 10 −70 1 −3 −3 40 0 −1 −4 1

(−1)R3−−−−−→

1 0 9 10 −70 1 −3 −3 40 0 1 4 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −26 20 1 0 9 10 0 1 4 −1

Solution 1 2 3 4 12 5 3 5 61 0 8 6 −6

R2−2R1−−−−−−→R3−R1

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

−→

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

R1−2R2−−−−−−→R3+2R2

1 0 9 10 −70 1 −3 −3 40 0 −1 −4 1

(−1)R3−−−−−→

1 0 9 10 −70 1 −3 −3 40 0 1 4 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −26 20 1 0 9 10 0 1 4 −1

Solution 1 2 3 4 12 5 3 5 61 0 8 6 −6

R2−2R1−−−−−−→R3−R1

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

−→

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

R1−2R2−−−−−−→R3+2R2

1 0 9 10 −70 1 −3 −3 40 0 −1 −4 1

(−1)R3−−−−−→

1 0 9 10 −70 1 −3 −3 40 0 1 4 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −26 20 1 0 9 10 0 1 4 −1

Solution 1 2 3 4 12 5 3 5 61 0 8 6 −6

R2−2R1−−−−−−→R3−R1

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

−→

1 2 3 4 10 1 −3 −3 40 −2 5 2 −7

R1−2R2−−−−−−→R3+2R2

1 0 9 10 −70 1 −3 −3 40 0 −1 −4 1

(−1)R3−−−−−→

1 0 9 10 −70 1 −3 −3 40 0 1 4 −1

R1−9R3−−−−−−→R2+3R3

1 0 0 −26 20 1 0 9 10 0 1 4 −1

Theorem (2.3.3)

Let A be a square matrix.

1 If B is a square matrix satisfying BA = I, then B = A−1.

2 If B is a square matrix satisfying AB = I, then B = A−1.

Theorem (2.3.4)

Let A be an n× n matrix. Then TFAE.

1 A is invertible.

2 Ax = 0 has only trivial solution.

3 The RREF of A is In.

5 Ax = b is consistent for every n× 1 matrix b.

6 Ax = b has exactly one solution for every n× 1 matrix b.

Theorem (2.3.5)

Let A and B be square matrices of the same size. If AB is invertible, then Aand B must also be invertible.

Theorem (2.3.4)

Let A be an n× n matrix. Then TFAE.

1 A is invertible.

2 Ax = 0 has only trivial solution.

3 The RREF of A is In.

5 Ax = b is consistent for every n× 1 matrix b.

6 Ax = b has exactly one solution for every n× 1 matrix b.

Theorem (2.3.5)

Let A and B be square matrices of the same size. If AB is invertible, then Aand B must also be invertible.

Example (2.3.3)

What conditions must a, b and c satisfy in order for the system of equationsx1 + x2 + 2x3 = a

x1 + x3 = b

2x1 + x2 + 3x3 = c

to be consistent ?

1 1 2 a1 0 1 b2 1 3 c

Example (2.3.3)

What conditions must a, b and c satisfy in order for the system of equationsx1 + x2 + 2x3 = a

x1 + x3 = b

2x1 + x2 + 3x3 = c

to be consistent ?

1 1 2 a1 0 1 b2 1 3 c

Solution 1 1 2 a1 0 1 b2 1 3 c

R2−R1−−−−−−→R3−2R1

1 1 2 a0 −1 −1 b− a0 −1 −1 c− 2a

(−1)R2−−−−−→

1 1 2 a0 1 1 a− b0 −1 −1 c− 2a

R3+R2−−−−−→

1 1 2 a0 1 1 a− b0 0 0 c− a− b

Thus, the linear system is consistent if c− a− b = 0 or c = a+ b. #

Solution 1 1 2 a1 0 1 b2 1 3 c

R2−R1−−−−−−→R3−2R1

1 1 2 a0 −1 −1 b− a0 −1 −1 c− 2a

(−1)R2−−−−−→

1 1 2 a0 1 1 a− b0 −1 −1 c− 2a

R3+R2−−−−−→

1 1 2 a0 1 1 a− b0 0 0 c− a− b

Solution 1 1 2 a1 0 1 b2 1 3 c

R2−R1−−−−−−→R3−2R1

1 1 2 a0 −1 −1 b− a0 −1 −1 c− 2a

(−1)R2−−−−−→

1 1 2 a0 1 1 a− b0 −1 −1 c− 2a

R3+R2−−−−−→

1 1 2 a0 1 1 a− b0 0 0 c− a− b

Solution 1 1 2 a1 0 1 b2 1 3 c

R2−R1−−−−−−→R3−2R1

1 1 2 a0 −1 −1 b− a0 −1 −1 c− 2a

(−1)R2−−−−−→

1 1 2 a0 1 1 a− b0 −1 −1 c− 2a

R3+R2−−−−−→

1 1 2 a0 1 1 a− b0 0 0 c− a− b

Solution 1 1 2 a1 0 1 b2 1 3 c

R2−R1−−−−−−→R3−2R1

1 1 2 a0 −1 −1 b− a0 −1 −1 c− 2a

(−1)R2−−−−−→

1 1 2 a0 1 1 a− b0 −1 −1 c− 2a

R3+R2−−−−−→

1 1 2 a0 1 1 a− b0 0 0 c− a− b

Example (2.3.4)

What conditions must a, b and c satisfy in order for the system of equationsx1 + 2x2 + 3x3 = a

2x1 + 5x2 + 3x3 = b

x1 + 8x3 = c

to be consistent ?

1 2 3 a2 5 3 b1 0 8 c

Example (2.3.4)

What conditions must a, b and c satisfy in order for the system of equationsx1 + 2x2 + 3x3 = a

2x1 + 5x2 + 3x3 = b

x1 + 8x3 = c

to be consistent ?

1 2 3 a2 5 3 b1 0 8 c

Solution 1 2 3 a2 5 3 b1 0 8 c

R2−2R1−−−−−−→R3−R1

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

−→

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

R1−2R2−−−−−−→R3+2R2

1 0 9 5a− 2b0 1 −3 b− 2a0 0 −1 c+ 2b− 5a

(−1)R3−−−−−→

1 0 9 5a− 2b0 1 −3 b− 2a0 0 1 5a− 2b− c

R1−9R3−−−−−−→R2+3R3

1 0 0 −40a+ 16b+ 9c0 1 0 13a− 5b− 3c0 0 1 5a− 2b− c

Thus, x1 = −40a+ 16b+ 9c, x2 = 13a− 5b− 3c, x3 = 5a− 2b− c is the solution wherea, b, c ∈ R. #

Solution 1 2 3 a2 5 3 b1 0 8 c

R2−2R1−−−−−−→R3−R1

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

−→

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

R1−2R2−−−−−−→R3+2R2

1 0 9 5a− 2b0 1 −3 b− 2a0 0 −1 c+ 2b− 5a

(−1)R3−−−−−→

1 0 9 5a− 2b0 1 −3 b− 2a0 0 1 5a− 2b− c

R1−9R3−−−−−−→R2+3R3

1 0 0 −40a+ 16b+ 9c0 1 0 13a− 5b− 3c0 0 1 5a− 2b− c

Solution 1 2 3 a2 5 3 b1 0 8 c

R2−2R1−−−−−−→R3−R1

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

−→

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

R1−2R2−−−−−−→R3+2R2

1 0 9 5a− 2b0 1 −3 b− 2a0 0 −1 c+ 2b− 5a

(−1)R3−−−−−→

1 0 9 5a− 2b0 1 −3 b− 2a0 0 1 5a− 2b− c

R1−9R3−−−−−−→R2+3R3

1 0 0 −40a+ 16b+ 9c0 1 0 13a− 5b− 3c0 0 1 5a− 2b− c

Solution 1 2 3 a2 5 3 b1 0 8 c

R2−2R1−−−−−−→R3−R1

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

−→

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

R1−2R2−−−−−−→R3+2R2

1 0 9 5a− 2b0 1 −3 b− 2a0 0 −1 c+ 2b− 5a

(−1)R3−−−−−→

1 0 9 5a− 2b0 1 −3 b− 2a0 0 1 5a− 2b− c

R1−9R3−−−−−−→R2+3R3

1 0 0 −40a+ 16b+ 9c0 1 0 13a− 5b− 3c0 0 1 5a− 2b− c

Solution 1 2 3 a2 5 3 b1 0 8 c

R2−2R1−−−−−−→R3−R1

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

−→

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

R1−2R2−−−−−−→R3+2R2

1 0 9 5a− 2b0 1 −3 b− 2a0 0 −1 c+ 2b− 5a

(−1)R3−−−−−→

1 0 9 5a− 2b0 1 −3 b− 2a0 0 1 5a− 2b− c

R1−9R3−−−−−−→R2+3R3

1 0 0 −40a+ 16b+ 9c0 1 0 13a− 5b− 3c0 0 1 5a− 2b− c

Solution 1 2 3 a2 5 3 b1 0 8 c

R2−2R1−−−−−−→R3−R1

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

−→

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

R1−2R2−−−−−−→R3+2R2

1 0 9 5a− 2b0 1 −3 b− 2a0 0 −1 c+ 2b− 5a

(−1)R3−−−−−→

1 0 9 5a− 2b0 1 −3 b− 2a0 0 1 5a− 2b− c

R1−9R3−−−−−−→R2+3R3

1 0 0 −40a+ 16b+ 9c0 1 0 13a− 5b− 3c0 0 1 5a− 2b− c

Solution 1 2 3 a2 5 3 b1 0 8 c

R2−2R1−−−−−−→R3−R1

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

−→

1 2 3 a0 1 −3 b− 2a0 −2 5 c− a

R1−2R2−−−−−−→R3+2R2

1 0 9 5a− 2b0 1 −3 b− 2a0 0 −1 c+ 2b− 5a

(−1)R3−−−−−→

1 0 9 5a− 2b0 1 −3 b− 2a0 0 1 5a− 2b− c

R1−9R3−−−−−−→R2+3R3

1 0 0 −40a+ 16b+ 9c0 1 0 13a− 5b− 3c0 0 1 5a− 2b− c

Assignment 2

1. In each part use the information to find A

(a) (EVEN) (5AT + 2I)−1 =

[8 55 3

](b) (ODD) (A+ 3I)−1 =

[9 −295 −16

2. Solve the linear system by x = A−1b.

(a) (EVEN)

x1 + 2x2 + 3x3 = 1

x2 + 4x3 = 2

5x1 + 6x2 = 3

(b) (ODD)

x2 + 7x3 = 1

x1 + 3x2 + 3x3 = 2

−2x1 − 5x2 = 3

3. (EVEN and ODD) What conditions must a, b and c satisfy in order for the system ofequations

2x1 + x2 + 3x3 + x4 = a

x1 + x3 + 2x4 = b

x1 + x2 + 2x3 + 4x4 = c

to be consistent ?

MAT2305 LINEAR ALGEBRA :Week2 · Outline for Week 2 Chapter 2 Matrices 2.1 Matrices and Matrix...

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