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