Question

The inverse of matrixis? $\left|\begin{array}{ccc}1& 0& 1\\ -1& 1& 1\\ 0& 1& 0\end{array}\right|$

Answer 1

$Cof\left(A\right)=\left|\begin{array}{ccc}-1& 0& -1\\ 1& 0& -1\\ -1& -2& 1\end{array}\right|$
& $\left|A\right|=1(0-1+0+1(-1-0)=-2\ne 0so{A}^{-1}exist$
So $adj=(CofA{)}^T=\left[\begin{array}{ccc}-1& 1& -1\\ 0& 0& -2\\ -1& -1& 1\end{array}\right]$
$A^{-1}=\frac{adj\left(A\right)}{\left|A\right|}=\frac{1}{(-2)}\left[\begin{array}{ccc}-1& 0& -1\\ 1& 0& -1\\ -1& -2& 1\end{array}\right]$
$=\frac{1}{2}\left[\begin{array}{ccc}1& -1& 1\\ 0& 0& 2\\ 1& 1& -1\end{array}\right]$