Exercise 13 Page 202

Two matrices are inverses of each other if their product is the identity matrix.

Practice makes perfect

We want to determine if K and L are inverse matrices. K= [ cc 1 & 2 3 & 0 ] L= [ cc 0 & 1 2 & -1 ]Two matrices are inverses of each other if their product is the identity matrix. MatricesA andB are inverses ⇕ A * B = B * A = I, where I= [ cc 1 & 0 0 & 1 ] Recall that the Commutative Property of Multiplication does not hold for matrices. Therefore, for K and L to be inverses, both K* L and L* K have to equal the identity matrix. Let's start by calculating K* L.

1 & 2 3 & 0 * 0 & 1 2 & -1 ? = 1 & 0 0 & 1

▼

Simplify left-hand side

MultMatrices

Multiply matrices

1(0) +2(2) & 1(1)+2(-1) 3(0)+0(2) & 3(1)+0(-1) ? = 1 & 0 0 & 1

Multiply

0+4 & 1-2 0+0 & 3-0 ? = 1 & 0 0 & 1

AddSubTerms

Add and subtract terms

0 & -1 0 & 3 ≠ 1 & 0 0 & 1 *

Since K* L ≠ I, we do not need to check if L* K is equal to I. We already know that matrices K and L are not inverses.