4.5, 4.62 x 2 and 3 x 3 Matrices, Determinants, and

Inverses

Date: _____________

Matrices are multiplicative inverses

• Page 199 – 2 definitions• Multiplicative Identity Matrix– Must be a square matrix, 2 x 2, 3 x 3, 4 x 4, etc. – Has 1’s in the main diagonal and 0’s elsewhere

• Multiplicative Inverse of a Matrix– when multiplying a matrix by its inverse, we get the

identity matrix1  A A I 1  A A I

1 0 01 0

  , 0 1 00 1

0 0 1

Use your calculator

Matrices are multiplicative inverses

2 5 8 5  ,

3 8 3 2

2 5 

3 8A

8 5 

3 2B

Show that these two matrices are multiplicative inverses

2 5 8 5 

3 8 3 2A B

1 0 

0 1

Objective - To evaluate the determinates of 2 x 2 and 3 x 3 matrices.

2 2 Matrix

2 6  det

9 7

2 6

9 72(7)

2 6

9 7

Determinant can be labeled either way

Find the Determinant

2 2 Matrix

2 6 det 

9 7

2 6

9 72(7)

2 6

9 7

Determinant

  9(6)

14 54

40a b

c dad cb

Objective - To evaluate the determinates of 2 x 2 and 3 x 3 matrices.

Find the Determinant

Evaluate the Determinant for each Matrix

1 31.  

5 2

3 72.  

2 9

1(2) 3( 5) 13

2 23.  

2 2

3(9) ( 2)7 41

( 2)(2) ( 2)(2) 0When the determinant = 0, then that matrix has NO INVERSE

 

4 2 5

   8 1 6

2 5 1

4 2 5

8 1 6

2 5 1

 

4

8

2

  

2

1

5

DeterminantTake the first 2 columns and rewrite them outside

Find the determinant of each 3x3 Matrix.

2 5 4 2

8 6  8  

4

12 5 2

1 1

5

4

Find the determinant of each 3x3 Matrix.

4 5 4 2

8 1  8  1

2 25

2

6

1 5

244

4 2 4 2

8 1 6     1

2 5 1 2

5

8

5

(        )200 244

4 2 4 2

8 6  8  

5

1

2

1

5 1 2 5

0(1(        )200 244

4 2 5 2

8 1  8  1

2 2

4

6

5 1 5

120 0(1(        )200 244

4 2 5 4

8 1 6     1

2 5 1 2

2

8

5

)16

228 146 82

    Solve

1 3 2

2 4 6  

3 1 9

120 0(1(        )200 244

Fun?Use your Calculator

1 3 2

2 4 6

3 1 9

A

Matrix, over to MATH, then det(, then go to Matrix, we want matrix A

det( ) 86A

Determinant and its useThe determinant is used to find our inverseWe will use our calculator to find the inverse. Type in:

2 3

1 1A

2 3

det1 1

Find the determinant first:

(2)(1) (3)(1) 1

Therefore, it has an inverse

Determinant and its useThe determinant is used to find our inverseWe will use our calculator to find the inverse. Type in:

2 3

1 1A

1A

1 3

1 2

3 2

11

2

A 1 4

2 6

Find the inverse of the matrix

1A

1 2 1

1.5 3 1.75

0 1 0.5

A

1 0 2

3 2 1

6 4 0

1A

If A didn’t have an inverse, you’d get the message ERR: SINGULAR MAT

Checking your answers.

AA 1  1 0

0 1

If you multiply inverses, you will always get the identity matrix.

This is a way you can check your answers

Solve for X.

Linear EquationsMatrix Equations

AX Bax b a a

bx

a

1 1A AX A B 1IX A B

1X A B

Objective - To solve systems using inverse matrices.

4 6 1

3 7 2X

4 6

3 7A

1

2B

1X A B1.9

1.1

Do this one on your own to see if you understand

6 10 13 84

4 2 7 18

0 9 8 56

X

5

8

2

6 10 13

4 2 7

0 9 8

A

84

18

56

B

1X A B