Determinants and Determinants and Multiplicative Inverses Multiplicative Inverses

of Matricesof MatricesSection 2-5

Before finishing this section you should be able to:

• Evaluate determinants• Find inverses of matrices• Solve systems of equations by

using inverses of matrices

Remember: Your textbook is your friend! This presentation is just a supplement to the text. BEFORE you view this, make sure you read this section in your textbook and look at all the great examples that are also worked there for you.

DeterminantsDeterminants

0 2

8 6

0 2

8 6

0 2

8 6

Each square matrix has a determinant. The determinant of

is a number denoted by

or det

.

The determinant of a 2 x 2 matrix is the difference of the products of the diagonals.

a b

c d ad bc -

4 5

3 8

4 5

3 8Q

Find the value of = 0(-6) - 8(-2) = 0 + 16 = 16

If , find the value of det(Q).

= (-4)(-8) – (5)(3) = 32 – 15 = 17

0 2

8 6

Finding the determinant of a Finding the determinant of a 3x3 matrix3x3 matrix

The minor of an element of any nth-order determinant is a determinant of order (n-1). We take the elements in the FIRST ROW and find their minors. The minor can be found by deleting the row and column containing the element.

For instance, find the determinant of 5 3 1

6 4 8

0 3 7

5 3 14 8 6 8 6 4

6 4 8 5 3 ( 1)3 7 0 7 0 3

0 3 7

This is called expansion by minors.

Notice that this is SUBTRACTED – the second term is always subtracted

Notice that this is ADDED – the third term is always added

Simplify and expand.

5[4(7)-(-3)(8)] - 3[6(7)-0(8)] + (-1)[6(-3)-0(4)]

= 5[28+24] -3[42-0] + (-1)[-18-0]

= 5[52]-3[42]+(-1)[-18] = 260-126+18 =152

5 3 14 8 6 8 6 4

6 4 8 5 3 ( 1)3 7 0 7 0 3

0 3 7

Another ExampleAnother Example

2 -3 -5

1 2 2

5 3 -1

2 22

3 1

1 2

( 3)5 1

1 2( 5)

5 3

Find the value of

.

= 2(-8) + 3(-11) – 5(-7)

= -14

Identity MatrixIdentity Matrix

1 0 0

0 1 0

0 0 1

2 3 1 0 2 3

1 8 0 1 1 8

The identity matrix for multiplication is a square matrix whose elements in the main diagonal, from upper left to lower right, are 1’s, and all other elements are 0’s.

Any square matrix A multiplied by the identity matrix, I , will equal A.

The identity matrix can also be 3 x 3, 4 x 4, 5 x 5, whatever square dimensions you need for that particular problem.

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Finding the Inverse of a Finding the Inverse of a matrixmatrix

An inverse matrix, A-1 , is a matrix that, when multiplied by matrix A, produces the identity matrix.

An inverse matrix of a 2 x 2 matrix can be found by multiplying a matrix by the reciprocal of the determinant, switching the 2 numbers of the main diagonal, and changing the signs of the other diagonal,

from top right to bottom left. 8 9

3 1

8 98( 1) 3(9) 35

3 1

The reciprocal of the determinant is multiplied by the altered version of the original matrix. (the elements on the main diagonal switch places, the elements on the other diagonal change signs).

An inverse exists only if the determinant is not equal to zero !!!!!!!!!!!!

For example,Find the inverse of the matrix

First find the determinant  

1 91

3 835

1 9

35 353 8

35 35

The inverse of the matrix is

Another ExampleAnother Example

4 2

3 2

4 2

3 2

4 2

3 2

2 21

3 42

1 1

32

2

Find the inverse of the matrix

.

First, find the determinant of

.

= 4(2) - 3(2) or 2

The inverse is

Finding the Inverse of a 3 x Finding the Inverse of a 3 x 3 matrix3 matrix

We will use our calculator to find this. It is pretty messy by hand. First, enter your matrix.Go to 2nd, then MATRIX

Hit the x-1 key

Go to MATRIX, Names and hit ENTER

To change the decimals to fractions hit MATH then ENTER, then ENTER again

Scroll over to EDIT and hit ENTEREnter the matrix

2 2 4

3 7 3

5 0 8

Solving a system of equations using Solving a system of equations using matricesmatrices

1 1

1

1

AX B

A AX A B

IX A B

X A B

2 3 17

1 1 4

x

y

Write the system as a matrix equation.

You can use a matrix inverse to solve a matrix equation in the form AX = B. To solve this equation for X, multiply each side of the equation by the inverse of A.

Solve the system of equations by using matrix equations.2x + 3y = -17 x – y = 4

1 3

5 51 2

5 5

1 3 1 32 3 175 5 5 5

1 2 1 1 1 2 4

5 5 5 5

x

y

1

5

x

y

Multiply both sides of the matrix equation by the inverse of the coefficient matrix.

 To solve the matrix equation, first find the inverse of the coefficient matrix which is

1 31 0 175 50 1 1 2 4

5 5

x

y

Example #2: solving a system of equations Example #2: solving a system of equations using matricesusing matrices

3 2 3

2 4 2

x

y

4 2 4 2 31 13 2 2 3 2 3 216

2 4

4 2 3 21

2 3 2 416

x

y

4 2 31

2 3 216

x

y

1

0

Solve the system of equations by using matrix equations.3x + 2y = 32x – 4y = 2

Write the system as a matrix equation.

To solve the matrix equation, first find the inverse of the coefficient matrix.

Now multiply each side of the matrix equation by the inverse and solve.

The solution is (1, 0).

Real-world exampleReal-world exampleBANKING A teller at Security Bank received a deposit from a local retailer containing only twenty-dollar bills and fifty-dollar bills. He received a total of 70 bills, and the amount of the deposit was $3200.

How many bills of each value were deposited?

First, let x represent the number of twenty-dollar bills and let y represent the number of fifty-dollar bills.

So, x + y = 70 since a total of 70 bills were deposited.

Write an equation in standard form that represents the total amount deposited.

20x + 50y = 3,200

Now solve the system of equations x + y = 70 and 20x + 50y = 3200. Write the system as a matrix equation and solve.

x + y = 70 20x + 50y = 3200

1 1

20 50

x

y

70

3200

Multiply each side of the equation by the inverse of the coefficient matrix.

50 1 1 11

20 1 20 5030

x

y

50 1 701

20 1 320030

x

y

10

60

The solution is (10, 60). The deposit contained 10 twenty-dollar

bills and 60 fifty-dollar bills.

Helpful WebsitesHelpful WebsitesFinding Determinants/Inverses of Matrices:http://www.purplemath.com/modules/determs.htmhttp://www.mathcentre.ac.uk/resources/leaflets/firstaidkits/5_4.pdf

Solving systems using matrices:http://math.uww.edu/faculty/mcfarlat/matrix.htm

2-5 self-Check Quiz: http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-07-860861-9&chapter=2&lesson=5&quizType=1&headerFile=4&state=