Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 8 - Section 8.4 - Multiplicative Inverses of Matrices and Matrix Equations - Exercise Set - Page 934: 85


The provided statement is false.

Work Step by Step

The matrix $AX=B$ is determined if we consider the simultaneous equations. $\begin{align} & x+2y=4 \\ & 3x-5y=1 \\ \end{align}$ Now, write in matrix form as below, Where, $A=\left[ \begin{matrix} 1 & 2 \\ 3 & -5 \\ \end{matrix} \right]$ $X=\left[ \begin{align} & X \\ & Y \\ \end{align} \right]$ $B=\left[ \begin{align} & 4 \\ & 1 \\ \end{align} \right]$ Now, solve the matrix $AX=B$ For X, multiply $A$ and the inverse of $B$, which is not possible. So, the statement is false. There is a formula to find the $X$, $AX=B$. Multiply both sides by the inverse of $A$. ${{A}^{-1}}AX={{A}^{-1}}B$ Where ${{A}^{-1}}A=I$ which is the identity matrix. Therefore $IX=X$ So, $X={{A}^{-1}}B$. There is a formula for finding the value of $X$. Then, the expression is true. If A is invertible, it is not possible to solve the equation for X.
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