4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using...

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4.5 2x2 Matrices, Determinants and Inverses 1. Evaluating Determinants of 2x2 Matrices 2. Using Inverse Matrices to Solve Equations

Transcript of 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using...

Page 1: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

4.52x2 Matrices, Determinants and

Inverses1. Evaluating Determinants of 2x2 Matrices

2. Using Inverse Matrices to Solve Equations

Page 2: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

When you multiply two matrices together, in the order AB or BA, and the result is the identity matrix, then matrices A and B are inverses.

10

01I

Identity matrix

Page 3: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

To show two matrices are inverses…

AB = I OR BA = I

AA-1 = I OR A-1A = I

Inverse of A Inverse of A

You only have to prove ONE of these.

Page 4: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Show that B is the multiplicative inverse of A.

17

13A

3.07.0

1.01.0B

Page 5: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Show that B is the multiplicative inverse of A.

17

13A

3.07.0

1.01.0B

3.07.0

1.01.0

17

13AB

Page 6: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Show that B is the multiplicative inverse of A.

17

13A

3.07.0

1.01.0B

3.07.0

1.01.0

17

13AB

10

01AB

AB = I. Therefore, B is the inverse of A and A is the inverse of B.

Page 7: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Show that B is the multiplicative inverse of A.

17

13A

3.07.0

1.01.0B

3.07.0

1.01.0

17

13AB

17

13

3.07.0

1.01.0BA

10

01AB

Check by multiplying BA…answer should be the same

AB = I. Therefore, B is the inverse of A and A is the inverse of B.

Page 8: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Show that B is the multiplicative inverse of A.

17

13A

3.07.0

1.01.0B

3.07.0

1.01.0

17

13AB

17

13

3.07.0

1.01.0BA

10

01AB

10

01BA

Check by multiplying BA…answer should be the same

AB = I. Therefore, B is the inverse of A and A is the inverse of B.

Page 9: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 2:

Show that the matrices are multiplicative inverses.

83

52A

23

58B

Page 10: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 2:

Show that the matrices are multiplicative inverses.

83

52A

23

58B

83

52

23

58BA

10

01BA

BA = I. Therefore, B is the inverse of A and A is the inverse of B.

Page 11: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

The determinant is used to tell us if an inverse exists.

If det ≠ 0, an inverse exists.

If det = 0, no inverse exists.

1) Evaluating Determinants of 2x2 Matrices

Page 12: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

To calculate a determinant…

dc

baA dc

baA det

Page 13: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

To calculate a determinant…

dc

baA dc

baA det

dc

ba Multiply along the diagonal

Page 14: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

To calculate a determinant…

dc

baA dc

baA det

dc

ba

bcad

Multiply along the diagonal

Equation to find the determinant

Page 15: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

95

87det

Page 16: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

95

87det

95

87det

Page 17: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

95

87det

95

87

95

87det

Page 18: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

95

87det

95

87

)5)(8()9)(7(

23

det = -23

Therefore, there is an inverse.

95

87det

Page 19: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 2: Evaluate the determinant.

24

24det

Page 20: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 2: Evaluate the determinant.

24

24det

)2)(4()2)(4( 0

24

24det

Page 21: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 2: Evaluate the determinant.

24

24det

)2)(4()2)(4( 0

24

24det

det = 0

Therefore, there is no inverse.

Page 22: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

How do you know if a matrix has an inverse AND what that inverse is?

ac

bd

AA

det

11

ac

bd

bcadA

11

Equations to find an inverse matrix

p.201

Page 23: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Page 24: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Step 1: Find det M

Page 25: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Step 1: Find det M

)5)(2()4)(2( bcad

2

det M = -2, the inverse of M exists.

Page 26: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Step 2: Rewrite the matrix in form.

ac

bd

Page 27: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Change signs

Step 2: Rewrite the matrix in form.

ac

bd

Page 28: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Change signs

?5

2?

Step 2: Rewrite the matrix in form.

ac

bd

Page 29: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Change positions

?5

2?

Step 2: Rewrite the matrix in form.

ac

bd

Page 30: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Step 2: Rewrite the matrix in form.

ac

bd

25

24

Change positions

Page 31: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Step 3: Use the equation to find the inverse.

25

24

25

24

2

11M

Page 32: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 1:

Determine whether the matrix has an inverse. If an inverse exists, find it.

45

22M

Step 3: Use the equation to find the inverse.

25

24

25

24

2

11M

15.2

121M

Page 33: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 2:

Determine whether the matrix has an inverse. If an inverse exists, find it.

31

42

Page 34: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 2:

Determine whether the matrix has an inverse. If an inverse exists, find it.

31

42

)1)(4()3)(2( bcad

2

31

42

31

42det

Page 35: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

1) Evaluating Determinants of 2x2 Matrices

Example 2:

Determine whether the matrix has an inverse. If an inverse exists, find it.

31

42

21

43

2

11A

15.0

25.11A

Page 36: 4.5 2x2 Matrices, Determinants and Inverses 1.Evaluating Determinants of 2x2 Matrices 2.Using Inverse Matrices to Solve Equations.

Homework

p.203 #1, 2, 4, 5, 14, 15, 27, 28, 32, 34