## Algebra and Trigonometry 10th Edition

Published by Cengage Learning

# Chapter 10 - 10.3 - The Inverse of a Square Matrix - 10.3 Exercises - Page 734: 11

#### Answer

B is the inverse of A.

#### Work Step by Step

When there is an inverse of a matrix $A A^{-1} = I$ (the Identity Matrix) $AB = \begin{bmatrix} 2 & 0 & 2 &1 \\3 & 0 & 0 & 1 \\ -1 &1 & -2 & 1 \\ 3 &-1 &1 &0 \end{bmatrix} \dfrac{1}{3}\begin{bmatrix} -1 & 3 & -2 & -2\\-2 & 9 &-7 &-10 \\ 1 & 0 & -1 &-1 \\ 3 & -6 & 6 & 6 \end{bmatrix}$ Now, multiplying by columns, we have: $AB = \dfrac{1}{3}\begin{bmatrix} 3 & 0 & 0 & 0\\0 & 3 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{bmatrix} =\begin{bmatrix} 1 & 0 & 0 & 0\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &1 \end{bmatrix}$ Since $AB=BA$ equals the identity matrix, B is the inverse of A.

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