To find the inverse of a matrix $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$, by adjoint matrix method.

$Adj$ of $a_{11}=\left|\begin{array}{cc}-3& 4\\ -1& 1\end{array}\right|=1$

$Adj$ of $a_{12}=-\left|\begin{array}{cc}2& 4\\ 0& 1\end{array}\right|=-2$

$Adj$ of $a_{13}=+\left|\begin{array}{cc}2& 4\\ -1& 1\end{array}\right|=-2$

$Adj$ of $a_{21}=-\left|\begin{array}{cc}-3& 4\\ -1& 1\end{array}\right|=-1$

$Adj$ of $a_{31}=\left|\begin{array}{cc}-3& 4\\ -3& 4\end{array}\right|=0$

$Adj$ of $a_{32}=-\left|\begin{array}{cc}3& 4\\ 2& 4\end{array}\right|=-4$

$Adj$ of $a_{22}=\left|\begin{array}{cc}3& 4\\ 0& 1\end{array}\right|=3$

$Adj$ of $a_{23}=-\left|\begin{array}{cc}3& -3\\ 0& -1\end{array}\right|=3$

$Adj$ of $a_{33}=\left|\begin{array}{cc}3& -3\\ 2& -3\end{array}\right|=3$

$|A=3(-3+4)-2(-3+4)+0=1$

$A^{\prime }=\frac{AdjA}{|A|}=\left[\begin{array}{ccc}1& -2& -2\\ -1& 3& 3\\ 0& -4& 3\end{array}\right]$