Answer
.
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]$
