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**4.5 2x2 Matrices, Determinants and Inverses**

Evaluating Determinants of 2x2 Matrices Using Inverse Matrices to Solve Equations

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**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. Identity matrix

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**Evaluating Determinants of 2x2 Matrices**

You only have to prove ONE of these. To show two matrices are inverses… AB = I OR BA = I AA-1 = I OR A-1A = I Inverse of A Inverse of A

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**Evaluating Determinants of 2x2 Matrices**

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

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**Evaluating Determinants of 2x2 Matrices**

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

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Show that B is the multiplicative inverse of A. AB = I. Therefore, B is the inverse of A and A is the inverse of B.

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Show that B is the multiplicative inverse of A. AB = I. Therefore, B is the inverse of A and A is the inverse of B. Check by multiplying BA…answer should be the same

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Show that B is the multiplicative inverse of A. AB = I. Therefore, B is the inverse of A and A is the inverse of B. Check by multiplying BA…answer should be the same

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**Evaluating Determinants of 2x2 Matrices**

Example 2: Show that the matrices are multiplicative inverses.

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**Evaluating Determinants of 2x2 Matrices**

Example 2: Show that the matrices are multiplicative inverses. BA = I. Therefore, B is the inverse of A and A is the inverse of B.

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**Evaluating Determinants of 2x2 Matrices**

The determinant is used to tell us if an inverse exists. If det ≠ 0, an inverse exists. If det = 0, no inverse exists.

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**Evaluating Determinants of 2x2 Matrices**

To calculate a determinant…

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**Evaluating Determinants of 2x2 Matrices**

To calculate a determinant… Multiply along the diagonal

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**Evaluating Determinants of 2x2 Matrices**

To calculate a determinant… Multiply along the diagonal Equation to find the determinant

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Evaluate the determinant.

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Evaluate the determinant.

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Evaluate the determinant.

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Evaluate the determinant. det = -23 Therefore, there is an inverse.

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**Evaluating Determinants of 2x2 Matrices**

Example 2: Evaluate the determinant.

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**Evaluating Determinants of 2x2 Matrices**

Example 2: Evaluate the determinant.

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**Evaluating Determinants of 2x2 Matrices**

Example 2: Evaluate the determinant. det = 0 Therefore, there is no inverse.

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**Evaluating Determinants of 2x2 Matrices**

How do you know if a matrix has an inverse AND what that inverse is? Equations to find an inverse matrix p.201

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**Evaluating Determinants of 2x2 Matrices**

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

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**Evaluating Determinants of 2x2 Matrices**

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

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Determine whether the matrix has an inverse. If an inverse exists, find it. Step 1: Find det M det M = -2, the inverse of M exists.

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form.

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form. Change signs

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form. Change signs

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form. Change positions

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form. Change positions

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Determine whether the matrix has an inverse. If an inverse exists, find it. Step 3: Use the equation to find the inverse.

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**Evaluating Determinants of 2x2 Matrices**

Example 1: Determine whether the matrix has an inverse. If an inverse exists, find it. Step 3: Use the equation to find the inverse.

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**Evaluating Determinants of 2x2 Matrices**

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

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**Evaluating Determinants of 2x2 Matrices**

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

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**Evaluating Determinants of 2x2 Matrices**

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

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

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2.5 Determinants and Multiplicative Inverses of Matrices. Objectives: 1.Evaluate determinants. 2.Find the inverses of matrices. 3.Solve systems of equations.

2.5 Determinants and Multiplicative Inverses of Matrices. Objectives: 1.Evaluate determinants. 2.Find the inverses of matrices. 3.Solve systems of equations.

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