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: 42

Answer

b) The value of the inverse matrix is $ X=\left\{ \left( 2,1,-1,3 \right) \right\}$
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Work Step by Step

a) Consider the given system of equations: $\begin{align} & 2w+2y+z=6 \\ & 3w+z=9 \\ & -w+x-2y+z=4 \\ & 4w-x+y=6 \end{align}$ The linear system can be written as: $ AX=B $ Where, $ A=\left[ \begin{matrix} 2 & 0 & 1 & 1 \\ 3 & 0 & 0 & 1 \\ -1 & 1 & -2 & 1 \\ 4 & -1 & 1 & 0 \\ \end{matrix} \right]$ $ X=\left[ \begin{align} & w \\ & x \\ & y \\ & z \\ \end{align} \right]$ $ B=\left[ \begin{align} & 6 \\ & 9 \\ & 4 \\ & 6 \\ \end{align} \right]$ (b) Consider the given system of equations: $\begin{align} & 2w+y+z=6 \\ & 3w+z=9 \\ & -w+x-2y+z=4 \\ & 4w-x+y=6 \end{align}$ The linear system can be written as: $ AX=B $ Where, $ A=\left[ \begin{matrix} 2 & 0 & 1 & 1 \\ 3 & 0 & 0 & 1 \\ -1 & 1 & -2 & 1 \\ 4 & -1 & 1 & 0 \\ \end{matrix} \right]$ $ X=\left[ \begin{align} & w \\ & x \\ & y \\ & z \\ \end{align} \right]$ $ B=\left[ \begin{align} & 6 \\ & 9 \\ & 4 \\ & 6 \\ \end{align} \right]$ Now, we will consider the coefficient matrix is $ A=\left[ \begin{matrix} 2 & 0 & 1 & 1 \\ 3 & 0 & 0 & 1 \\ -1 & 1 & -2 & 1 \\ 4 & -1 & 1 & 0 \\ \end{matrix} \right]$ Use the inverse of the coefficient matrix to get ${{\left[ A \right]}^{-1}}=\left[ \begin{matrix} -1 & 2 & -1 & -1 \\ -4 & 9 & -5 & -6 \\ 0 & 1 & -1 & -1 \\ 3 & -5 & 3 & 3 \\ \end{matrix} \right]$ Now, to find the values of the provided system: we will use the formula $ X={{A}^{-1}}B $ Where, ${{\left[ A \right]}^{-1}}=\left[ \begin{matrix} -1 & 2 & -1 & -1 \\ -4 & 9 & -5 & -6 \\ 0 & 1 & -1 & -1 \\ 3 & -5 & 3 & 3 \\ \end{matrix} \right]$ $ B=\left[ \begin{align} & 6 \\ & 9 \\ & 4 \\ & 6 \\ \end{align} \right]$ Now, substitute the values in $ X={{A}^{-1}}B $
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