Finite Math and Applied Calculus (6th Edition)

Published by Brooks Cole
ISBN 10: 1133607705
ISBN 13: 978-1-13360-770-0

Chapter 4 - Section 4.3 - Matrix Inversion - Exercises - Page 263: 6

Answer

$A$ and $B$ are inverses of each other.

Work Step by Step

The inverse of an $n\times n$ matrix $A$ is that $n\times n$ matrix $A^{-1}$ which, when multiplied by $A$ on either side, yields the $n\times n$ identity matrix $I$. $A A^{-1}=A^{-1}A=I.$ --- $aa^{-1}=bb^{-1}=cc^{-1}=1,$ $AB=\left[\begin{array}{lll} 1+0+0 & 0+0+0 & 0+0+0\\ 0+0+0 & 0+1+0 & 0+0+0\\ 0+0+0 & 0+0+0 & 0+0+1 \end{array}\right]=\left[\begin{array}{lll} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right]=I_{3}$ $BA=\left[\begin{array}{lll} 1+0+0 & 0+0+0 & 0+0+0\\ 0+0+0 & 0+1+0 & 0+0+0\\ 0+0+0 & 0+0+0 & 0+0+1 \end{array}\right]=\left[\begin{array}{lll} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right]=I_{3}$ $A$ and $B$ are inverses of each other.
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