2x2 matrices, determinants and inverses

2x2 Matrices, Determinants and Inverses

1. Evaluating Determinants of 2x2 Matrices2. 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.

1001

I

Identity matrix for multiplication

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.

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

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

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.

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.

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.

1) Evaluating Determinants of 2x2 Matrices

Example 2:Show that the matrices are multiplicative inverses.

8352

A

2358

B

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.

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

1) Evaluating Determinants of 2x2 Matrices

To calculate a determinant…

dcba

A dcba

A det

1) Evaluating Determinants of 2x2 Matrices

To calculate a determinant…

dcba

A dcba

A det

dcba

Multiply along the diagonal

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

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

9587

det

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

9587

det

9587

det

1) Evaluating Determinants of 2x2 Matrices

Example 1: Evaluate the determinant.

9587

det

9587

9587

det

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

1) Evaluating Determinants of 2x2 Matrices

Example 2: Evaluate the determinant.

2424

det

1) Evaluating Determinants of 2x2 Matrices

Example 2: Evaluate the determinant.

2424

det

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

2424

det

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.

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

1) Evaluating Determinants of 2x2 Matrices

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

4522

M

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

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.

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

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

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

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

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

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

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

1) Evaluating Determinants of 2x2 Matrices

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

3142

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

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