2x2 Matrices, Determinants and Inverses

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

description

2x2 Matrices, Determinants and Inverses. Evaluating Determinants of 2x2 Matrices Using Inverse Matrices to Solve Equations. Evaluating Determinants of 2x2 Matrices. - PowerPoint PPT Presentation

Transcript of 2x2 Matrices, Determinants and Inverses

Page 1: 2x2 Matrices, Determinants and Inverses

2x2 Matrices, Determinants and Inverses

1. Evaluating Determinants of 2x2 Matrices2. Using Inverse Matrices to Solve Equations

Page 2: 2x2 Matrices, Determinants and Inverses

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.

1001

I

Identity matrix for multiplication

Page 3: 2x2 Matrices, Determinants and Inverses

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: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Show that B is the multiplicative inverse of A.

1713

A

3.07.01.01.0

B

Page 5: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Show that B is the multiplicative inverse of A.

1713

A

3.07.01.01.0

B

3.07.01.01.0

1713

AB

Page 6: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Show that B is the multiplicative inverse of A.

1713

A

3.07.01.01.0

B

3.07.01.01.0

1713

AB

1001

AB

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

Page 7: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Show that B is the multiplicative inverse of A.

1713

A

3.07.01.01.0

B

3.07.01.01.0

1713

AB

1713

3.07.01.01.0

BA

1001

AB

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: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Show that B is the multiplicative inverse of A.

1713

A

3.07.01.01.0

B

3.07.01.01.0

1713

AB

1713

3.07.01.01.0

BA

1001

AB

1001

BA

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: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 2:Show that the matrices are multiplicative inverses.

8352

A

2358

B

Page 10: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 2:Show that the matrices are multiplicative inverses.

8352

A

2358

B

8352

2358

BA

1001

BA

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

Page 11: 2x2 Matrices, Determinants and Inverses

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

If det ≠ 0, an inverse exists.

If det = 0, no inverse exists. A Matrix with a determinant of zero is called a SINGULAR matrix

1) Evaluating Determinants of 2x2 Matrices

Page 12: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

To calculate a determinant…

dcba

A dcba

A det

Page 13: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

To calculate a determinant…

dcba

A dcba

A det

dcba

Multiply along the diagonal

Page 14: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

To calculate a determinant…

dcba

A dcba

A det

dcba

bcad

Take the product of the leading diagonal, and subtract the product of the non-leading diagonal

Equation to find the determinant

Page 15: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

9587

det

Page 16: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

9587

det

9587

det

Page 17: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

9587

det

9587

9587

det

Page 18: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

9587

det

9587

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

23

det = -23Therefore, there is an inverse.

9587

det

Page 19: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 2: Evaluate the determinant.

2424

det

Page 20: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 2: Evaluate the determinant.

2424

det

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

2424

det

Page 21: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 2: Evaluate the determinant.

2424

det

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

2424

det

det = 0

Therefore, there is no inverse.

Page 22: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

How do you know if a matrix has an inverse AND what that inverse is?Given , the inverse of A is given by:

acbd

AA

det11

Equation to find an inverse matrix

This is called the adjoint matrix. It is formed by interchanging elements in the leading diagonal and negating elements in the non-leading diagonal

dcba

A

Page 23: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

M

Page 24: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

MStep 1: Find det M

Page 25: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

MStep 1: Find det M

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

2

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

Page 26: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

MStep 2: Find the adjoint matrix. i.e

acbd

Page 27: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

MChange signs

Step 2: Find the adjoint matrix. i.e

acbd

Page 28: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

MChange signs

?52?

Step 2: Find the adjoint matrix. i.e

acbd

Adjoint of M

Page 29: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

MChange positions

?52?

Step 2: Find the adjoint matrix. i.e

acbd

Adjoint of M

Page 30: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

MStep 2: Find the adjoint matrix. i.e

acbd

2524

Change positions

Adjoint of M

Page 31: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

MStep 3: Use the equation to find the inverse.

2524

211M

ofMAdjoM

M intdet

11

Page 32: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 1:Determine whether the matrix has an inverse. If an inverse exists, find it.

4522

MStep 3: Use the equation to find the inverse.

2524

211M

15.2121M

Page 33: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 2:Determine whether the matrix has an inverse. If an inverse exists, find it.

3142

Page 34: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 2:Determine whether the matrix has an inverse. If an inverse exists, find it.

3142

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

2

3142

3142

det

Page 35: 2x2 Matrices, Determinants and Inverses

1) Evaluating Determinants of 2x2 Matrices

Example 2:Determine whether the matrix has an inverse. If an inverse exists, find it.

3142

2143

211A

15.025.11A