Answer
(a) $x=1$, $y=1$, $z=-1$.
(b) $x=0$, $y=1$, $z=-1$.
Work Step by Step
(a) The coefficient matrix $A$ is given by
$$A=\left[ \begin {array}{ccc} 1&2&1\\ 1&2&-1
\\ 1&-2&1\end {array} \right]
.$$
Using Gauss-Jordan elimination, one can calculate $A^{-1}$ as follows
$$A^{-1}=\left[ \begin {array}{ccc} 0&\frac{1}{2}&\frac{1}{2}\\ \frac{1}{4}&0&-\frac{1}{4}
\\ \frac{1}{2}&- \frac{1}{2}&0\end {array} \right]
.
$$
The solution is given by
$$\left[ \begin {array}{cc} {x}\\ {y} \\{z}\end {array} \right]=\left[ \begin {array}{ccc} 0&\frac{1}{2}&\frac{1}{2}\\ \frac{1}{4}&0&-\frac{1}{4}
\\ \frac{1}{2}&- \frac{1}{2}&0\end {array} \right]\left[ \begin {array}{cc} {2}\\ {4}\\{-2} \end {array} \right]=\left[ \begin {array}{c} 1\\ 1\\-1\end {array}
\right].
$$
That is, $x=1$, $y=1$, $z=-1$.
The coefficient matrix $A$ is given by
$$A=\left[ \begin {array}{ccc} 1&2&1\\ 1&2&-1
\\ 1&-2&1\end {array} \right]
.$$
Using Gauss-Jordan elimination, one can calculate $A^{-1}$ as follows
$$A^{-1}=\left[ \begin {array}{ccc} 0&\frac{1}{2}&\frac{1}{2}\\ \frac{1}{4}&0&-\frac{1}{4}
\\ \frac{1}{2}&- \frac{1}{2}&0\end {array} \right]
.
$$
The solution is given by
$$\left[ \begin {array}{cc} {x}\\ {y}\\{z} \end {array} \right]=\left[ \begin {array}{ccc} 0&\frac{1}{2}&\frac{1}{2}\\ \frac{1}{4}&0&-\frac{1}{4}
\\ \frac{1}{2}&- \frac{1}{2}&0\end {array} \right]\left[ \begin {array}{cc} {1}\\ {3}\\{-3} \end {array} \right]=\left[ \begin {array}{c} 0\\ 1\\-1\end {array}
\right].
$$
That is, $x=0$, $y=1$, $z=-1$.
