12.4 Inverses of Matrices

Remember if A and B are inverses, AB = I and BA = I *only square matrices can have multiplicative inverses* Ex 1) Show that matrix B is the multiplicative inverse of matrix A.

To find the inverse of a square 2 × 2 matrix, we: (1)Find the determinant ad – bc (2)Make some changes in your matrix: (3)Multiply Ex 2) Find the multiplicative inverse. det = 27 – 28 = –1 switch change sign

For 3 × 3 and higher, we can use a calculator! Ex 3) Find the multiplicative inverse. 2 nd MATRIX  EDIT  [A] 3 × 3 enter data QUIT 2 nd MATRIX  choose [A] [A] –1 = yikes! change to fractions MATH 1: ►Frac *arrow over to see the rest*

We can use inverses to solve for an unknown matrix *Be careful of the order* If A, X, and B are matrices, and AX = B to “get rid” of A, we multiply by A –1 A –1 (AX) = A –1 B X = A –1 B (must be in this order!) Ex 4) Solve for X.

Ex 5) Solve for X.(Use your calculator!) ↑ enter for matrix A ↑ enter for matrix B X = A –1 B

We can take a system of equations and turn it into a matrix equation to then solve! Ex 6) Set up the matrices to solve the system (We’re not going to solve this one – just set it up. But you solve in the homework!) coefficients ↑ represents answers You continue to solve like the previous examples

Homework #1204 Pg 624 #1–13 odd, 16, 21, 23, 31, 39, 41