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 933: 41

Answer

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1570422332

Work Step by Step

(b) Consider the given system of equations, $\begin{align} & w-x+2y=-3 \\ & x-y+z=4 \\ & -w+x-y+2z=2 \\ & -x+y-2z=-4 \end{align}$ The linear system can be written as: $ AX=B $ Where, $ A=\left[ \begin{matrix} 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \\ \end{matrix} \right]$ $ X=\left[ \begin{align} & w \\ & x \\ & y \\ & z \\ \end{align} \right]$ $ B=\left[ \begin{align} & -3 \\ & 4 \\ & 2 \\ & -4 \\ \end{align} \right]$ Now, consider the coefficient matrix $ A=\left[ \begin{matrix} 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \\ \end{matrix} \right]$ Use the inverse of the coefficient matrix, to get, ${{\left[ A \right]}^{-1}}=\left[ \begin{matrix} 0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1 \\ \end{matrix} \right]$ Now, to get the value of the provided system we will use the formula $ X={{A}^{-1}}B $ Where, ${{\left[ A \right]}^{-1}}=\left[ \begin{matrix} 0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1 \\ \end{matrix} \right]$ $ B=\left[ \begin{align} & -3 \\ & 4 \\ & 2 \\ & -4 \\ \end{align} \right]$
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