If $\left[\begin{array}{ccc}1& 1& 1\\ 1& -1& 1\\ 2& 1& -1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}6\\ 2\\ 1\end{array}\right]$  then

Let $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& -1& 1\\ 2& 1& -1\end{array}\right]$  then $\left|A\right|=1\left(0\right)-\left(-3\right)+1\left(3\right)=6$ 

 $adjA=\left[\begin{array}{ccc}0& 2& 2\\ 3& -3& 0\\ 3& 1& -2\end{array}\right]$  

Now $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=A^{-1}B=\left[\begin{array}{ccc}0& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{2}& -\frac{1}{2}& 0\\ \frac{1}{2}& \frac{1}{6}& -\frac{1}{3}\end{array}\right]\left[\begin{array}{c}6\\ 2\\ 1\end{array}\right]$ 

 $=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\Rightarrow x=1,y=2,z=3$