3.8 – Use Inverse Matrices to Solve Linear Systems The n x n identity matrix is a matrix with 1’s on the main diagonal and 0’s elsewhere. If A is any n x n matrix and I is the n x n identity matrix, then AI = A and IA = A.

3.8 – Use Inverse Matrices to Solve Linear Systems Two n x n matrices A and B are inverses of each other if their product (in both orders) is the n x n identity matrix. That is AB = I and BA = I. An n x n matrix A has an inverse if and only if det A does not = 0. The symbol for the inverse of A is A -1.

3.8 – Use Inverse Matrices to Solve Linear Systems Example 1: Find the inverse of the matrix.

3.8 – Use Inverse Matrices to Solve Linear Systems Example 2: Solve the matrix equation AX = B for the 2 x 2 matrix X.

3.8 – Use Inverse Matrices to Solve Linear Systems

Example 3: Use an inverse matrix to solve the linear system.

3.8 – Use Inverse Matrices to Solve Linear Systems Example 4: Use an inverse matrix to solve the linear system. -2x + 3y = -11 5x + y = 19

3.8 – Use Inverse Matrices to Solve Linear Systems Example 5: Use an inverse matrix to solve the linear system. 4x - y = 10 -7x – 2y = -25

3.8 – Use Inverse Matrices to Solve Linear Systems Inverse of a 3 x 3 matrix The inverse of a 3 x3 matrix is difficult to compute by hand…use your calculator to compute this inverse.

3.8 – Use Inverse Matrices to Solve Linear Systems Example 6: Use a graphing calculator to find the inverse of A. Then use the calculator to verify your results.

3.8 – Use Inverse Matrices to Solve Linear Systems Example 7: Use a graphing calculator to find the inverse of A. Then use the calculator to verify your results.

3.8 – Use Inverse Matrices to Solve Linear Systems Example 8: A company sells three types of movie gift baskets. A basic basket with 2 movie passes and 1 package of microwave popcorn costs $ A medium basket with 2 movie passes, 2 packages of popcorn, and 1 DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVD’s costs $ Find the cost of each in the gift baskets.