1 MAC 2103 Module 3 Determinants. 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Determine the minor, cofactor,

1

MAC 2103

Module 3

Determinants

2Rev.F09

Learning Objectives

Upon completing this module, you should be able to:

1. Determine the minor, cofactor, and adjoint of a matrix.

2. Evaluate the determinant of a matrix by cofactor expansion.

3. Determine the inverse of a matrix using the adjoint.

4. Solve a linear system using Cramer’s Rule.

5. Use row reduction to evaluate a determinant.

6. Use determinants to test for invertibility.

7. Find the eigenvalues and eigenvectors of a matrix.

http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

http://faculty.valenciacc.edu/ashaw/

3Rev.09

Determinants


Determinants by Cofactor ExpansionDeterminants by Cofactor Expansion

Evaluating Determinants by Row ReductionEvaluating Determinants by Row Reduction

Properties of the DeterminantProperties of the Determinant

There are three major topics in this module:


4Rev.F09

What is a Determinant?


Determinants are commonly used to test if a matrix is invertible and to find the area of certain geometric figures.

A determinant is a real number associated with a square matrix.

= a bc d


5Rev.F09

How to Determine if a Matrix is Invertible?


The following is often used to determine if a square matrix is invertible.


6Rev.F09

Example


Determine if A-1 exists by computing the determinant of the matrix A.

a) b)

Solution

a)

b)

AA-1-1 does not exist does not exist

AA-1-1 does exist does exist

det(A)= 9 3−3 −1

=(9)(−1)−(−3)(3) =0

det(A)= −5 94 −1

=(−5)(−1)−(4)(9) =−31


7Rev.F09

What are Minors and Cofactors?


We know we can find the determinants of 2 x 2 matrices; but can we find the determinants of 3 x 3 matrices, 4 x 4 matrices, 5 x 5 matrices, ...?

In order to find the determinants of larger square matrices, we need to understand the concept of minors and cofactors.


8Rev.F09

Example of Finding Minors and Cofactors


Find the minor M11 and cofactor A11

for matrix A.

Solution

To obtain M11 begin by crossing out the first row and

column of A.The The minorminor is equal to is equal to det det B = B = −−6(5) 6(5) −− ( (−−3)(7) 3)(7) = = −−99Since Since AA1111 = ( = (−−1)1)1+11+1MM1111, , AA1111 can can

be computed as follows:be computed as follows:

AA1111 = ( = (−−1)1)22((−−9) = 9) = −−9 9


9Rev.F09

How to Find the Determinant of Any Square Matrix?


Once we know how to obtain a cofactor, we can find the determinant of any square matrix. You may pick any row or column, but the calculation is easier if some elements in the selected row or column equal 0.

aijAiji=1

n

∑for any column j

aijAijj=1

n

∑for any row i

or


10

Rev.F09

Example of Finding the Determinant by Cofactor Expansion


Find det A, if

Solution To find the determinant of A, we can select any row or column. If we begin expanding about the first column of A, then

det A = a11

A11

+ a21

A21

+ a31

A31

.

A11

= −9 from the previous example

A21

= −12 and A31

= 24

det A = a11

A11

+ a21

A21

+ a31

A31

= (−8)(−9) + (4)(−12) + (2)(24)

= 72

Now, try to find the determinant of A by expanding the first row of A.


11

Rev.F09

How to Find the Adjoint of a Matrix?


The adjoint of a matrix can be found by taking the transpose of the matrix of cofactors from A.

In our previous example, we have found the cofactors A11, A21, A31. If we continue to solve for the rest of the cofactors for matrix A, namely A12, A22, A32 , A13, A23, and A33 , then we will have a 3 x 3 matrix of cofactors from A as follows:

A11 A12 A13A21 A22 A23A31 A32 A33

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


12

Rev.F09

How to Find the Adjoint of a Matrix? (Cont.)


The transpose of this 3 x 3 matrix of cofactors from A is called the adjoint of A, and it is denoted by Adj(A).

What are we going to do with this Adj(A)? We can use it to help us find the A-1 if A is an invertible matrix.

Adj(A)=

A11 A21 A31

A12 A22 A32

A13 A23 A33

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

13

Rev.F09

How to Find A-1 Using the Adjoint of a Matrix?


Theorem 2.1.2: If A is an invertible matrix, then

A−1 =1

det(A)Adj(A)

Note:

1. The square matrix A is invertible if and only if det(A) is not zero.

2. If A is an n x n triangular matrix, then det(A) is the product of the entries on the main diagonal of the matrix (Theorem 2.1.3.)

14

Rev.F09

What is Cramer’s Rule?


Cramer’s Rule is a method that utilizes determinants to solve systems of linear equations. This rule can be extended to a system of n linear equations in n unknowns as long as the determinant of the matrix is non-zero.


15

Rev.F09

Example of Using Cramer’s Rule to Solve the Linear System


Use Cramer’s rule to solve

the linear system.

Solution In this system a1 = 1, b1 = 4, c1 = 3, a2 = 2, b2

= 9 and c2 = 5


16

Rev.F09

Example of Using Cramer’s Rule to Solve the Linear System (cont.)


E = 7, F = −1 and D = 1

The solution is

Note that Gaussian elimination with backward substitution is usually more efficient than Cramer’s Rule.


17

Rev.F09

What Are the Limitations on the Method of Cofactors and Cramer’s Rule?

The main limitations are as follow:

1. A substantial number of arithmetic operations are needed to compute determinants of large matrices.

• The cofactor method of calculating the determinant of an n x n matrix, n > 2, generally involves more than n! multiplication operations.

• Time and cost required to solve linear systems that involve thousands of equations in real-life applications.

Next, we are going to look at a more efficient method to find the determinant of a general square matrix.


18

Rev.F09

Evaluating Determinants by Reducing the Matrix to Row-Echelon Form


Let A be a square matrix. (See Theorem 2.2.3)

(a) If B is the matrix that results from scaling by a scalar k, then

det(B) = k det(A).

(b) If B is the matrix that results from either rows interchange or columns interchange, then

det(B) = - det(A).

(c) If B is the matrix that results from row replacement, then

det(B) = det(A).

Just keep these in mind when A is a square matrix:

1. det(A)=det(AT).

2. If A has a row of zeros or a column of zeros, then det(A)=0.

3. If A has two proportional rows or two proportional columns, then det(A)=0.


19

Rev.F09

How to Evaluate the Determinant by Row Reduction?


Let’s look at a square matrix A.

We can find the determinant by reducing it into row-echelon form.

Step 1: We want a leading 1 in row 1. We can interchange row 1 and row 2 to accomplish this.

A =0 3 11 1 23 2 4

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥


20

Rev.F09

How to Evaluate the Determinant by Row Reduction? (Cont.)


Step 2: We want a leading 1 in row 2. We can take a common factor of 3 from row 2 to accomplish this (scaling).

Step 3: We want a zero at both row 2 and row 3 below the leading 1 in row 1. We can add -3 times row 1 to row 3 to accomplish this (row replacement).

det(A)=−1 1 20 3 13 2 4

From Step 1:

det(A)=−31 1 20 1 1

3

0 −1 −2

det(A)=−31 1 20 1 1

3

3 2 4

21

Rev.F09

How to Evaluate the Determinant by Row Reduction? (Cont.)


Step 4: We want a zero below the leading 1 in row 2. We can add row 2 to row 3 to accomplish this (row replacement).

Step 5: We want a leading 1 in row 3. We take a common factor of -5/3 from row 3 to accomplish this (scaling).

From Step 3:

det(A)=−31 1 20 1 1

3

0 −1 −2det(A)=−3

1 1 20 1 1

3

0 0 −53

det(A)=(−3)−53

⎛⎝⎜

⎞⎠⎟

1 1 20 1 1

3

0 0 1

=(−3)−53

⎛⎝⎜

⎞⎠⎟(1) =5

Remember: If A is an n x n triangular matrix, then det(A) is the product of the entries on the main diagonal of the matrix.

22

Rev.F09

Let’s Look at Some UsefulBasic Properties of Determinants


• Let A and B be n x n matrices and k is any scalar. Then,

• If A is invertible, then

This is because A-1A=I, det(A-1A) =det(I) =1; det(A-1) det(A) = 1, and so

det(kA)=kn det(A)det(AB)=det(A)det(B)

det(A−1) =1

det(A)

det(A−1) =1

det(A),det(A) ≠0.

Question:

Is det(A+B) = det(A) + det(B) ?

Remember: If A is an n x n triangular matrix, then det(A) is the product of the entries on the main diagonal of the matrix.

23

Rev.F09

What are Eigenvalues and EigenVectors?


An eigenvector of an n x n matrix A is a nontrivial (nonzero) vector such that , where is a scalar called an eigenvalue.

Linear systems of this form can be rewritten as follows:

The system has a nontrivial solution if and only if

This is the so called characteristic equation of A and therefore B has no inverse, and the linear system

has infinitely many solutions.

Arx =λ r

x rx λ

λ rx − A

rx =

r0

(λ I − A)rx = B

rx =

r0

det(λI −A) =det(B) =0. rx

Brx =

r0


24

Rev.F09

(λI −A)rx=

r0.


Express the following linear system in the form

Find the characteristic equation, eigenvalues

and eigenvectors corresponding to each of the eigenvalues.

The linear system can be written in matrix form as

with

x1 + 2x2 =λx1

2x1 + x2 =λx2

1 22 1

⎡

⎣⎢

⎤

⎦⎥

x1

x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=λ

x1

x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

A = 1 22 1

⎡

⎣⎢

⎤

⎦⎥,

rx =

x1

x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

λx1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥− 1 2

2 1

⎡

⎣⎢

⎤

⎦⎥

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 0

0

⎡

⎣⎢

⎤

⎦⎥

λ 1 00 1

⎡

⎣⎢

⎤

⎦⎥

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥− 1 2

2 1

⎡

⎣⎢

⎤

⎦⎥

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 0

0

⎡

⎣⎢

⎤

⎦⎥

Example

25

Rev.F09

(λI −A)rx=

r0.


λ 00 λ

⎡

⎣⎢

⎤

⎦⎥−

1 22 1

⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜

⎞

⎠⎟

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 0

0

⎡

⎣⎢

⎤

⎦⎥

λ −1 −2−2 λ −1

⎡

⎣⎢

⎤

⎦⎥

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 0

0

⎡

⎣⎢

⎤

⎦⎥

which is of the form

Thus,

Can you tell what is the characteristic equation for A?

λI − A = λ −1 −2−2 λ −1

⎡

⎣⎢

⎤

⎦⎥.

Example (Cont.)

26

Rev.F09http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

The characteristic equation for A is

or(λ −1)(λ −1)−(−2)(−2) =0

(λ −1)2 −4 =0

λ2 −2λ +1−4 =0

λ2 −2λ −3 =0(λ −3)(λ +1) =0

Example (Cont.)

27


Thus, the eigenvalues of A are:

By definition, is an eigenvector of A if and only if is a nontrivial solution of

that is

If , then we have

Thus, we can form the augmented matrix and solve by Gauss Jordan Elimination.

λ1 = 3,λ 2 = −1

Example (Cont.)

rx

rx

(λI −A)rx=

r0.

λ −1 −2−2 λ −1

⎡

⎣⎢

⎤

⎦⎥

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 0

0

⎡

⎣⎢

⎤

⎦⎥

λ =3

28


Let’s form the augmented matrix and solve by Gauss Jordan Elimination.

Thus,

a free variable,

Example (Cont.)

r1r2

2 −2−2 2

00

⎡

⎣⎢

⎤

⎦⎥

12 r1→ r1

r21 −1−2 2

00

⎡

⎣⎢

⎤

⎦⎥

r12r1+ r2 → r2

1 −10 0

00

⎡

⎣⎢

⎤

⎦⎥

x1 −x2 =0x1 =x2 =t t ∈(−∞,∞)

29


Solving this system yields:

So the eigenvectors corresponding to

are the nontrivial solutions of the form

Similarly, if , then we have

Example (Cont.)

λ1 = 3

−2 −2−2 −2

⎡

⎣⎢

⎤

⎦⎥

x1x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 0

0

⎡

⎣⎢

⎤

⎦⎥

−2x1 − 2x2−2x1 − 2x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= 0

0

⎡

⎣⎢

⎤

⎦⎥

x1 =tx2 =t

rx1 =

x1

x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= t

t⎡

⎣⎢

⎤

⎦⎥=t 1

1⎡

⎣⎢

⎤

⎦⎥

λ =−1

30


Example (Cont.)Let’s form the augmented matrix and solve by Gauss Jordan Elimination.

Thus,

r1r2

−2 −2−2 −2

00

⎡

⎣⎢

⎤

⎦⎥

−12 r1→ r1

r21 1−2 −2

00

⎡

⎣⎢

⎤

⎦⎥

r12r1+ r2 → r2

1 10 0

00

⎡

⎣⎢

⎤

⎦⎥

31


Solving this system yields:

So the eigenvectors corresponding to

are the nontrivial solutions of the form

Example (Cont.)x1 =tx2 =−t

rx2 =

x1

x2

⎡

⎣⎢⎢

⎤

⎦⎥⎥= t

−t⎡

⎣⎢

⎤

⎦⎥=t 1

−1⎡

⎣⎢

⎤

⎦⎥

λ2 = −1

32

Rev.F09

What have we learned?

We have learned to:

1. Determine the minor, cofactor, and adjoint of a matrix.

2. Evaluate the determinant of a matrix by cofactor expansion.

3. Determine the inverse of a matrix using the adjoint.

4. Solve a linear system using Cramer’s Rule.

5. Use row reduction to evaluate a determinant.

6. Use determinants to test for invertibility.

7. Find the eigenvalues and eigenvectors of a matrix.



33

Rev.F09

Credit

Some of these slides have been adapted/modified in part/whole from the text or slides of the following textbooks:

• Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition

• Rockswold, Gary: Precalculus with Modeling and Visualization, 3th Edition



Documents

1 MAC 2103 Module 3 Determinants. 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Determine the minor, cofactor,