4.5 2x2 Matrices, Determinants and Inverses

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

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

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

Evaluating Determinants of 2x2 Matrices Example 1: Show that B is the multiplicative inverse of A.

Evaluating Determinants of 2x2 Matrices Example 1: Show that B is the multiplicative inverse of A.

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.

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

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

Evaluating Determinants of 2x2 Matrices Example 2: Show that the matrices are multiplicative inverses.

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.

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.

Evaluating Determinants of 2x2 Matrices To calculate a determinant…

Evaluating Determinants of 2x2 Matrices To calculate a determinant… Multiply along the diagonal

Evaluating Determinants of 2x2 Matrices To calculate a determinant… Multiply along the diagonal Equation to find the determinant

Evaluating Determinants of 2x2 Matrices Example 1: Evaluate the determinant.

Evaluating Determinants of 2x2 Matrices Example 1: Evaluate the determinant.

Evaluating Determinants of 2x2 Matrices Example 1: Evaluate the determinant.

Evaluating Determinants of 2x2 Matrices Example 1: Evaluate the determinant. det = -23 Therefore, there is an inverse.

Evaluating Determinants of 2x2 Matrices Example 2: Evaluate the determinant.

Evaluating Determinants of 2x2 Matrices Example 2: Evaluate the determinant.

Evaluating Determinants of 2x2 Matrices Example 2: Evaluate the determinant. det = 0 Therefore, there is no inverse.

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

Evaluating Determinants of 2x2 Matrices Example 1: Determine whether the matrix has an inverse. If an inverse exists, find it.

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

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.

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.

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

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

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

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

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.

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.

Evaluating Determinants of 2x2 Matrices Example 2: Determine whether the matrix has an inverse. If an inverse exists, find it.

Evaluating Determinants of 2x2 Matrices Example 2: Determine whether the matrix has an inverse. If an inverse exists, find it.

Evaluating Determinants of 2x2 Matrices Example 2: Determine whether the matrix has an inverse. If an inverse exists, find it.

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