Matrix and Determinant

Halil Aydemir - 19.06.2011

Matrix and Determinant

matrix and matrices

rows and columns

square matrix

entries

main diagonal

diagonal matrix

scalar matrix

zero matrix

identfy matrix

triangular matrices

matrix addition

matrix multiplication

transpose

symetric matrix

skew-symetric matrix

reduced row echelon form

linear systems

gouss-jordon reduction

inverse of matrix

trace of matrix

determinant

cramer's rule

adjoint

rank

subspace

linear combination

spans

linearly independent/dependent

basis

dimension

eigenvalues , eigenvectors and diagonalization

characteristic polynomail

coyley–hamilton theorem


Matrix and Matrices

Definition: An mxn matrix A is a rectangular array of real number arranged in “m” ,

horizontally rows and “n” vertical columns.

A[ ]

- : the (i,j) so this mean entry of A.

-“m” and “n” are the integers.

-m: the number of rows of A

-n: the number of columns of A

*if m=n that A is called a square matrix.

Example:

A is a 3x3 matrix with entries; =1 , =0 ...

B is a 2x2 matrix with entries; =-1 , =0 ...

C is a 1x1 matrix with entries; =7

D is a 3x1 matrix with entries; =2 , =0 ...

Definition: Let A be an nxn matrix;

........... The entries , ..... are called

A= .......... the main diagonal entries.

. . .

. . .

.......... the main diagonal


Definition: Let A=[ ] be a square matrix if =0 for i j then A is called a diagonal

matrix.Its shape of a diagonal matrix is of the form;

...........

A= ..........

. . .

. . .

.......... the main diagonal

but

is square but not diagonal because of entries.

also

diag(0,0,0)=

or diag(2,-1,3)=

Definition: A diagonal marix A[ ] is called a scalar matrix if = ...= =c

(c is constant)

Example:

Definition: Let A=[ ] be an mxn matrix if all entries of A are zero than A is called zero

matrix.Zero matrix is donated by 0.

Example:

Definition: If all entries on the main diagonal are “1” than a diagonal matrix is an identfy

matrix.This matrix is donated by .

Example:

=

=


Definition: A square matrix A[ ] is said to be upper triangular if =0 for all i>j.İts shape

of an upper triangular matrix is of the form.

...........

A= ..........

0 . ..............

0 0 ...........

0 0..........

Definition: A square matrix A[ ] is said to be lower triangular if =0 for all i<j.İts shape

of an lower triangular matrix is of the form.

0...............0

A= .............0

. . ..............0

. . ............0

..........

*If the matrix is lower or upper matrix,it must be diagonal and square matrix.

Definition: Let A[ ] and B[ ] be two mxn matrices A and B are equal , if = for all

i,j.

Example: The matrices A=

and B=

so x=0,y=5 and z=1.

Matrix Addition

The sum of mxn matrices A=[ ] and B[ ] is define as the mxn matrix C[ ] with entries

= + for all i,j.The sum of two matrices A and B is defined only when A and B are the

same size.


Properties of Matrix Additions

Let A,B,C are mxn matrices;

1)A+B=B+A

2)A+(B+C)=(A+B)+C

3)For each mxn matrix A,there is a unique matrix B such that A+B=B+A=0 for mxn matrix.B

is donated by A and called the negative of A.

Multiplication by Scalars

Let A=[ ] be an mxn matrix.The matrix rA is defined as the mxn matrix B=[ ] with

entries.

Properties of Multiplication of Matrix

A,B matrices and r,s is real number;

1)r(A+B)=rA+rB

2)(r+s)=(rA).B=A(rB)

4)r(sA)=(rs)A=s(rA)

Definition: If , .... are mxn matrices and , ... are real numbers than

+ ....+ is called linear combination of , .... and , ... are called

coefficients.

Example:

and

then find the linear combination –A+3/2B.

-A+3/2B =

Matrix Multiplication

Definition: Let A be an mxp matrix and B be a pxn matrix.The product AB= the mxn

matrix given by = .

Note: If two matrices product,their size must be balance.

A=mxp

B=nxp so AB must be mxn size.


Example:

,

find AB and BA.

A . B B . A

2x3 = 3x3 A.B=

3x3 2x3 so B.A is undefined.

2x3

Example:

Let

and

so compute A.B

A.B

3x2 2x2

3x2 A.B=

Example:

Let

and

if A.B=

find “x” and “y”.

2x+2y+18=12 , x=1

4-y+6=6 , y=4

Example:

Let and

Find AB and BA.

and


Example:

Let

,

compute A.B

Example:

Let A =

, B=

and C=

compute A.B and A.C

A.B=

, A.C=

so A.B=A.C

Note: a,b,c R , a.b=a.c for a,b,c R and a 0 that b=c , but it’s not true fro matrices

A.B=A.C and A 0 whereas B C.

Properties of Matrix Multiplication:

If A,B,C are of the appropriate sizes then;

1) A(BC)=(AB)C

2) A(B+C)=AB+AC and (B+C)A=BA+CA

Remark: If A is an mxn matrix and then like;

A. =

Definition: If A=[ ] is an mxn matrix and the transpose of A is the nxm matrix

given for all i,j.

Example:

A=

, B= , C

= Compute .

,

,


Properties of Traspose:

1)

2)

3) r

4) !!!

If P and A is square matrix , we can define the powers of A as follows;

=A.A.A......A (p times)

Definition: If =A then A is symetric matrix.

If =-A then A is skew-symetric matrix.

1) A, are of the same size.

2) A is symetric if an only if for all i,j.

3) A is skew-symetric if an only if for all i,j

Definition: If an mxn matrix A satisfied the following properties then it is said to be in

reduced row echelon form.

a)All zero rows , if there are any appear at bottom of the matrix A.(bottom of the matrix must

be zero)

b)Each leading entry is one. (All rows must be start with 1)

c)For each nonzero row,the leading one appear to the right and below my leading one’s in

preceding.(other ones must be any right side of the first row one.)

d)Each leading one is the unique nonzero entry of its own column.(All columns , which has

leading , must be one or zero.)

Note: If the matrix provide first three part , it will called row echelon matrix.İf the matrix

provide all of the parts , it will called reduce row echelon matrix.

Example:

,

,


,

,

,

,

A is not row echelon form and also not in reduce row echelon form.

B is row echelon form and also reduce row echelon form .

C is row echelon form and also reduce row echelon form .

D is row echelon form and also reduce row echelon form .

E is not row echelon form and also not in reduce row echelon form.

F is not row echelon form and also not in reduce row echelon form.

G is row echelon form but not in reduce row echelon form.

H is not row echelon form and also not in reduce row echelon form.

T is row echelon form and also reduce row echelon form .

Definition: Any one of the following operations is called a elementary row operation on an

mxn matrix of A.

1) Interchange row î and row j.

2) Multiply row î by a nonzero scalar k.

3) Add k times row î to row j.(î j)

Definition: Let A and B be mxn matrices.A is row equivolen to B if B can be obtained by

applying a finite squence of elementary row operations to A.

Example:

and

conver A to B.

A~B

-2

2 -1

2


Theorem:

i)Every matrix is row equivalent to itself.(A~A)

ii)If A is row equivalent to B then B is row equivalent to A.(A~B so B~A)

iii)If A is row equivalent to B and B is row equivalent to C then A is now equivalent to

C.(A~B,B~C so A~C)

Note: Every mxn matrix is row equivalent to a unique matrix in reduced row echelon

form.This matrix is called the reduced row echelon form of the matrix.

Example:

Let

find the row echelon form of A.

~

~

row echelon and reduce row echelon

Linear System

A linear system of m equations in m unknowns

. . . .

. . . .

. . . .

, : constants

1

2

-2

1

2

-1

2


is called the coefficient matrix of the linear system.

,

this system can eb written in matrix form as follow;

AX=B

Gouss-Jordon Reduction

StepI : Form the augmented matrix [A|B].

Step II : Obtain the reduced row echelon form [C|D] of the augmented matrix

[A|B].

Step III : For each nonzero row,solve the corresponding equation for the

unknown associated with leading one in that row.

Definition: The augmented matrix is the mx(n+1) matrix it can be obtained by adjoining the

column B to A is donated by [A|B].

. . . .

. . . .

. . . .

[A|B]=

Example:

Solve the linear system x+2y+3z=9 by Gouss-Jordon reduction.

2x – y +z=8

3x -z =3


The augmented matrix is;

Step I :

Step II :

~

~

~

~

~

~

Step III : 1.x+0.y+0.z=2 so x=2

0.x+1.y+0.z=-1 so y=-1 and z=3

Example:

Solve the system x + y + 2z - 5w = 3 by Gouss-Jordon reduction.

2x + 5y - z - 9w= -3

2x + y - z + 3w = -11

x - 3y + 2z + 7w = -5

~

~

-2

-1/5

6

-1/4

-3

-1

-2

-1 1/4

-2

-7

-2


~

~

~

x + 2w = -5 so x = -5 – 2w

y – 3w = 2 so y = 2 +3w

z – 2w = 3 so z= 3+2w

Note: If the last column has a leading entry than the system has no solution.This is called

inconsisted.Otherwise it is called consisted.If system is constant and it has no free variable

then it has a unique solution.

Theorem: Let AX=B and CX=D so if [A|B] is row equivalent to [C|D] then AX=B and

CX=D have the same solution.Corollary , if A~C then AX=C and CX=0 have the same

solution.

Homogenius System

A linear system fo the form;

. . . .

. . . .

is called a homogenious system.A homogenius

system can be written in matrix form as AX=0 ,

-1/5

-2

-1

-2

4

-2

-2

-2

-1

-2


is a solution of the homogenius system.This solution is called

the trivial solution of the homogenius system.Other solutions of the given system are called

nontrivial solutions.

Example: Consider the homogenious system ;

~

~

~

~

so x=0 , y=0 , z=0 ; x=y=z is a unique solution of theorem.The given system has only trivial

solution.

Example: Solve the homogenious system;

~

~

~

~

x+w=0 , y-w=0 , z+w=0 so x=-w , y=w , z=-w , w =free so this mean so many solutions.

w=r in any real number.Thus the solution is x=-r , y=r , z=-r , w=r we can be assigned any

real number than the given system has infinitely many solutions and

general solution is

=

and solution set is

=

-2

1

-2

-2

-2

1/5 3

-1

-1

-1

-2 -1

-1

-1

-1

-1

-1

-1


Example: Find all solutions to the given linear system.

~

~

~

~

is in reduced row echelon form and are basic but is free variable;this system

has infinity many solution , , , , so;

the general solution is

and solution set is

Example: Find all solutions to the given linear system.

~

~

-3

-2

-1/2

-2

-1/2

-2

-2

-2 -2

-2

-1

-2

-2

-2

-3

-2

6

-2

+1

-2


are basic but is free so so

so

the general set is

and solution set is

Example: Solve the given system;

~

~

~

. so this system has no solution.

Example: Find conditions that the b’s satisfy for the system consistant;

~

the given system is consistant if and only if we have and

x2

-2 -3

-2

1/6

-2

-3

-2

8

-2

2

-2

-4

-2

3

-2

-1

-2

1

-2

-1

-2


Example: Find all values of a for which the resulting system has ;

a)no solution b)a unique solution c)infinitely many soltions

~

~

~

=0 then are critical point

if a=3 then

is free so th system has infinity many solution for a=3.

if a=-3 then

-6 so it has no solution for a=-3.

if then

~

so

given system has unique solution for

Example: Find all values of a for which the resulting system has ;

a)no solution b)a unique solution c)infinitely many soltions

-1

-2

-2 -1

-2

-1

-2

-1/2

-2


~

~

(a+2)(a-1)=0 so a=-2 and a=1 are critical point

if a=-2 then

0 3 it has no solution

if a=1 then

is basic but are free so the given system has infinity

many solutions.

if then

~

this system has unique solution.

Example: Let A and B symetric matrices;

a)show that A+B is symetric

b)show that A.B is symetric if and only if A.B=B.A

proof: , since A and B are symetric and if A+B symetric , A and B must be

equal to transpoze.

check that a)

A+B is symetric matrix.

b)

so

so A.B is symetric matrix

-1

-2

-2

-(1+a)

-2

-2

-a

-2

-2

-1

-2

-2

-2

-2

-2

-2


The Inverse of A Matrix

Let A be an mxn matrix and A is called invertible (non-singular) if there exist a matrix B

such that A.B = B.A = I. The matrix B is called the inverse of A .

Theorem: The inverse of a matrix if there exist is unique . Proof;

Support that B and C are inverses of A ( B=?C)

A.B =B.A =I and A.C=C.A=I since B and C are inverses of A so B=B.I=B.(A.C)=(B.A).C

=I.C=C then B=C.

Note: The inverse of A is donated by like A. = .A=I

Example: Let

find the .

A is invertible.

Note: For invertible matrix must be definitelly square matrix.

Remark: Not every matrix has an inverser.

Example:

find

Properties of the Inverse

a) If A is invertible then is invertible and

b) If A and B are invertible matrices then A.B is invertible and

c) If A is invertible then is invertible and .

Proof of a: since A is invertible , write A. = .A=I . We have to find a matrix B such

that .B=B. =I so B=A since the inverse is unique.Therefore,


Proof of b: It must be equal to right and left multiply. and such that

A. = .A=I and B. = .B=I .

From right multiply ; (A.B)( . )=A( .B) =A.I. =A. =I

From left multiply ; ( . )(A.B)= .A).B= .I.B=B. =I so they are equal.

Proof of c: From right ;

From left ;

the inverse of is so .

How to Find :

#Step1 : write [A|I]

#Step2 : obtain the reduced row echelon form [C|D] of [A|I].

#Step3 : if we obtain a zero row in the first part of matrix [C|D] then does not exist.If we

obtain an identify matrix I in the first part of matrix [C|D] then A is invertible and

=D.

Example:

[A|I]=

Note: [A|I] ~.....~.....~.....[C|D]

Example: [A|I]=

=?

does not exist invert , C is not invertible.

Definition: The trace of an nxn matrix A is desined by

Show that ;

a) where C is a scalar.

b)

c)

-1

-1

1/2

-3/2

-2


Proof of a : Let then so

Proof of b : Let then so then

Proof of c : so then so

so

Note: Tr(A.B)=Tr(B.A)

Example: Find the inverse of the given matrices of a,b,c but if they are possible.

a)

~

~

so

b)

~

non invertible because of zeros.

Determinant

Let be an nxn matrix. is the (n-1)x(n-1) submatrix obtained by deleting the

row and column of A. is called the minor of .

is called the cofactor of

expansion of |A| a long the row and same for

Example: If

-1

-1 -1

1/2

-1

3

-2


Example:

Example:

Example:

by using the first column expansion.

Example: If

fiind the determinant of te following matrices

Note: Change the place in determinant is having (-) .If multiply with some number of row

echolon , you must multiply with this number of determinant.If mutliply some row and add

other row , this calculate is not affect the determinant result.


Example: Compute the determinant

.


.


.


.

Example: Let

,

then

Example:

=

Example: Without expanding , show that

-4/5

1 1


Example: Without expanding , show that

Properties of Determinants

1) .

2) If A has a zero row or column then .

3) If B is obtained from A by multiplying a row or column of A by k then .

4) If two rows or column of A are equal then .

5) If B is obtained from A by interchancing two rows or columns of A then .

6) If B is obtained from A by adding to each elementary of the row or column of A a

constant k times corresponding element the row or column of A then .

7) If a matrix is upper or lower triangular then .

Corollary: The determinant of diagonal matrix is the product of entries on the main diagonal.

8) .

Corollary: If A is nonsingular(A is invertible) then and

.

Cramer’s Rule: Let A be an nxn matrix , Cramer’s rule for solving the linear system AX=B

is as follows :

#Step 1: Compute |A|. If |A|=0 then Cramer’s rule is not appliciable.Use Gauss-Jordan

reduction.

#Step 2: If |A| 0 then for each i ,

. is the matrix obtained from A by replacing the

column of A by B.

Example: Consider the following linear system ;

- 1 - 1


Example: |A| = 4 find | | and | |.

Example: If A & B are nxn matrices with |A| = 2 and |B| = -3 calculate | |.

Example:

without directly evulating.

Example: Prove that identify without evulating the determinants.

Note: Don’t do anything for calculating column or row.

+1

-1

+1

-1


Definition: Let A=[ ] be an Axn matrix then the matrix

is

callded the adjoint of A.

Example: Let

compute the adjoint of A.

Theorem: If A=[ ] is an nxn matrx then A(Adj(A))=(Adj(A)).A=|A|.

Corollary: If A is an nxn matrix and |A| 0 then

.

Proof: Adj(A) = (Adj(A))=|A|. & |A| 0.

since A has a unique solution.

Theorem: A matrix is nonsingular iff |A| 0.

Corollary: For an nxn matrix A , AX=0 has a nontrival solution iff |A|=0.

Example: If possible solve the following linear system by cramer’s rule.


Example: Use Cramer’s rule to find all values of a for which the linear system.

has the solution in which y=1.

Example: Prove the identify without evulating the determinants.

Example: Prove the identify without evulating the determinants.

Example: Without directly evulating , show that;

a

b c


Example: Find the determinant of

Example: The inverse of certain matrix A is given

use this

information to find |A| and Adj(A).

Example: Let

a) By using sutable elementary row and column operations as well as row and column

expansions , show that |A|=6.

b)Find (2,2) and (3,1) entries of .

-2

-2

1

1

1


a)

b)Transpose of (2,2) is (2,2) so

Transpose of (3,1) is (1,3) so

Note: When take determinant in adjoint , do not write entry!

Example: Find component “t” of the solution vector for following linear system.

-2

-1 -1

-1

-1

-2


Example: Without expanding , show that ;

Example: What must be “k” if the matrix

is not invertible.

If it is not invertible so it must be determinant=0

.

Example: Solve the system by using the inverse of the coefficient matrix

-1

-1

-1

-1

-2


Definition: A submatrix of A is any matrix obtained from A by deleting some rows , columns

of A , A is also considered to be a submatrix of A.

Definition: A nonzero matrix A is said to have rank r if atleast one of r-square submatrix is

nonsingular while every (r+1)-square submatrix of A is singular. A zero matrix is said to have

rank 0.

Example:

compute the rank of A.

How to Compute the Rank of Any Matrix

#Step1: Obtain the row echelon form B of A.

#Step2: The rank of A is the number of nonzero rows of B.

Some Knowledge about the Rank of a Matrix

Let A be an nxn matrix

1) A is nonsingular if and only if rank A=n.

2) Rank A=n if and only if

3) AX=B has a unique solution for every nx1 matrix B if and only if rank A=n.

4) AX=0 has a nontrivial solution if and only if rank A<n.

5) Elementery row operations don’t alter the rank of a matrix.

6) Equivalent matrices have the same rank.

-2

-3 -1


List of Nonsingular Equivalent

TFAE for an nxn matrix A

1) A is nonsingular

2)

3) Rank A = n

5) AX=0 has only trivial solution

6) AX=B has a unique solution.

7) A is row equivalent to I

Example:

compute the rank of A.

Rank of A is 3 and

Example:

compute the rank.

Rank is 3 and

Example:

compute the rank.

Rank is 2 and

-3

-2 -1/3

-1/4

-1

-1

-3

-2

-1

-2

1 -1


Subspace

Definition: An n-vector (or n-type) is an nx1 matrix.

row vector.

whose entries are real numbers called the components of x

: the set of all n-vectors.

: is called the n-spaces.

Definition: A no empty subset V of is a subspace of the following properties is satisfied.

1) If then .

2) If , then .

Note: 1)

2)

3)

Note: * , {0} are subspace of

* {0} is called the zero subspace of

* If V is a subspace then

Example: Consider the subset V of consisting of all vector of the form

, show

that V is a subspace of .

1)

2) Let

3) Let

1,2,3 is satisfied so subspace of .


Example: Consider the subset V of consisting of all vector of the form

.Is V a

subspace of ?

so V is

not subspace.

Note: If V is a subspace then 0 V.

Example: Let A be an mxn matrix and consider the linear system The set V of all

solutions to this system is a subset of .Then V is a subspace of

All s is true so this system is subspace of .

Definition: A vector V is is said to be linear combination of vector if it can

be written as where are constants (real numbers).

Example: Let

The vector V is a linear combination of

and .We must find constants.

,

1

-1

1/6

-2 -2


so V is a linear combination of , and V=2 .

Definition: Let S={ , } be set of vectors in a subspace V of . The set S spans

V or V is spanned by S if every vector in V is a linear combination of vector in S.

Example: Let S={ , } where

Determine whether S

spans .

X= ,

Example: Let S={ } where

Determine whether S spans

If then this system has no solution since a solution to this system can be

obtained for any choice of . Therefore , S does not span .

Definition: Set S={ } be a set of vectors in a subspace V of . The set S is

said to be linearly dependent , if we can find constants not a ll zero such that

0= . Otherwise S is linearly independent that is , S is linearly

independent if equation ca be satisfied only with

Example: Consider the vectors

Determine whether

S={ , } is linearly dependent or independent?

-2

-1

-1/3

2 3

1/3

-2

-2

-1


, so linearly independent.

Example: Consider this vectors

and

. Determine whether

S={ , } is linearly dependent or independent?

Thus homogenous system has nontrivial solutions.Therefore , S is lenearly dependent.

Theorem: Let A be an nxn matrix.Thn A is nonsingular iff the columns of A form a linearly

independent set S={ , } ; det(S)=0 this system is linearly dependent.Otherwise

linearly independent.

Definition: A set of vectors S={ ,..., } in a subspace V of is called a basis for V if

S spans V and S is linearly independent.

Note: If system have spans and linearly independent , this system called basis.

Example: The sets

and

are basis Determine the

basis for S and T.

i)

we find constants so S spans .

-2

-1

-1

-2 1/2

1

3 -1

-1


ii)

so linearly independent.

System S is spans and linearly independent so this system is basis.

i)

we find the constants so S spans

ii)


System S is spans and linearly independent so this system is basis.

Theorem: If S={ ,..., } is a basis for subspace V of and is a

linearly independent set of vector in V then

Corallary: S={ ,..., } and are basis for subspace V of then

.

Definition: The dimension of a subspace V of is the number of vectors in a basis for

V.Thus , dimension of is n.

dim( )=n

-1

-1

-1 -1/2

-1


dim( )=3

Definition: Suppose that S={ ,..., }is basis for a subspace V of and let .The

The vector

is called the coordinate vector of X with respect to S.

Example: Consider the basis

and

for .Let

Find .

for :

for :

Example: Which of the following subset are subspace of ?

i)

. ii)

iii)

i) First is not a subspace because of

but

ii) Second is a subspace because of ;

-1 -1 1

1

-1


*

*

iii) Third is not a subspace because of ;

Example: Let

determine if

belongs to spans?

so this system has o solution and

Example: Let

Determine if

belongs to spans.

Therefore ,

Example: Which of the following sets are linearly independent?

i)

ii)

iii)

i)

so this is linearly independent.

-1

1

-1

1/2

-1

-1

-1


ii)

so this system has many solution then linearly dependent.

iii)


Example: Consider the subset w of consisting all vectors of the form where a=0.

i) show that w is a subspace of

ii) find a basis for w

iii) find dimension of w

i)

* **Let

***Let

All is true so W is a subspace

ii) * Let

spans W.

**Determine whether S is a linearly independent


All is true so S is basis for W.

iii) dim(W)=2 because the number of vectors in S since S has 2 vector.

1

-2

-1


Example: Consider the subset

of

i)Show that V is a subspace of

ii)Find the basis for V

iii)Find dimension of V

i) *

**Let

so

***

All is true so V is a subspace of

ii) * Take

**

so S is linearly independent.

This equation is both linearly independent and spans.Therefore, S is basis for V.

*** dim(V)=3 b because of 3 vector.

Example: Find the set of vectors spanning the solution space of AX=0 where


and so ,

The solution set is

We know that the solution set is the space of AX=0 then;

spans for S.

Note: “AX=0” mean is same with “the null space of A”.

Definition: Let be a set of nonzero vectors in The procedure for

finding a subset of S that is a basis for W=Spans.

#Step1 Form the matrix having as its column vectors.

#Step2 Obtain the reduced row echelon form B of A.

#Step3 The vectors respond’ng to the columns containing the leading entry form a basis for

W=Spans.

Example: Let

Find a subset of S that is a basis for

W=Spans.

2

-1

4

1

1

-2

1

2

-1

1/6

3


has 2 leading entry so have 2 basis and dim(B)=2.

Note: “Find a subset of S that is a basis for W=Spans” mean is same with “Find a basis for

the subspace of spanned by S”.

Eigenvalues , Eigenvectors and Diagonalization

Definition: let A be an nxn matrix.The number λ is called an eigenvalues of A if there exists a

nonzero vector X in such that .

A nonzero vector X satisfying is called an eigenvector of A associated with the eigenvalue λ.

This system is a homogenous system.

is called the characteristic polynomial of A.

is called the characterisitic equation of A.

If A is an nxn matrix then is the characteristic polynomial of degree n.

Example: Let

Write characteristic polynomail of A and find eigenvalues

of A.

characteristic polynomial is

eigenvalues are

Theorem: An nxn matrix A is singular iff is an eigenvalue of A.

10

-1

-1


List of Nonsingular Eigenvalues

* A is nonsingular

*

* rank(A)=n

* A is row equivalent to

* is not an eigenvalue of A

* has only the trivial solution ( has a unique solution ).

Example: The eigenvalue of A are the roots of characteristic polynomial of A.Let

.Find eigenvalues of A and eigenvectors of A.

eigenvalues are

lets find eigenvectors of A;

, so

then

first eigenvector.

-4

-1 -1/2 1

-1/4 1 2

4


, so

then

second eigenvector.

, so

then

third eigenvector.

Definition: A matrix B is similar to A if there is a nonsingular matrix P so that

Some Properties:

* A is similar to A. (

* If B is similar to A then A is similar to B.

* If A is similar to B and b is similar to C then A is similar to C.

Definition: Let A be nxn matrix.A is diagonalizable if A is similar to a diagonal matrix D.

Also we can say that A ca be diagonolized.

Theorem: Similar matrices have the same eigenvalues. The eigenvalues of a diagonal matrix

are the entries on its ain diagonal.

Theorem: An nxn matrix a is diagonalizable iff A has an lineraly independent eigenvectors.

are three linearly independent eigenvectors and A is 3x3.Then A is diagonalizable.

Remark: If A is a diagonizable matrix then where D is a diagonal matrix. The

diagonal elements of D are eigenvalues of A. Moreover , P is matrix whose columns are n

linearly independent eigenvalues of A.

The order of the columns of p determines the order of diagonal elements in D.

4

1 1/2 1

-2


Example: Let

Compute the and determine A is

diagonalizable or not.

*

so

** because of the main diagonal

first eigenvector.

second eigenvector.

check that eigenvectors are linearly independent , if it is , A is diagonalizable.

This system is unique solution and constants are zero so linearly independent.Then A is

diagonalizable.

Example: Let

if possible ; find matrix P that diagonlizes A and determine

so

-1 -1

-2

1/2

-1

-1 -1

-1

-1

-1

-2


then is the eigenvector assicated with .

then is the eigenvector assicated with .

There don’t exist three linearly independent eigenvector of A.Therefore , A is not

diagonalizable.

.

Example: Let

if possible ; find matrix P that diagonlizes A and determine

.

so

, ,

then

is the eigenvector assicated with .

, ,

-1/2

-1/2

-1

2

-1/4

1

4

-4

-1/2

-1

1


then

is the eigenvector assicated with .

, ,

then

is the eigenvector assicated with

Therefore ; are linearly independent eigenvector since the rrots of characteristic

polynomial of A distrinct.A is diagonalizable.

Note: For writing , put the each value on the main diagonal.

Definition: Let A be an nxn matrix. Then the characteristic polynomial of A is

=

. The characteristic equation is

This equation has at most n distinct roots. A

has almost n distinct eiqenvalues for so

Example: Show that A and have the same eigenvalues.

Example: Let A be an nxn matrix ;

a)Show that |A| is the product of all the roots of the characteristic polynomial of A.

b)Show that A is singular if and only if is an eigenvalues of A.

a)

1/4

-2 -4

2

3


b) A is singular i.e. such that i.e. 0 is an

eigenvalues of A.

Coyley – Hamilton Theorem

If is the characteristic polynomial of A then

Example: Verify Coyley-Hamilton theorem for the following matrix

.

so

correct.

Example: Use the Coyley-Hamilton theorem to compute the inverse of

.

then

-1


Note: .This mean for using Coyley-Hamilton

thorem , determinant must be different than zero.

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Matrix and Determinant