Question

If $A=\left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 4\\ -2& 2& 1\end{array}\right]$ find $(A^{\prime }{)}^{-1}$

Answer 1

$A=\left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 4\\ -2& 2& 1\end{array}\right]$ $\to$ $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 1& 0\end{array}\right]$

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

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

$\left[\begin{array}{ccc}-1& -2& 3\\ 0& 1& 1\\ -1& 0& 4\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 0& 2& 1\end{array}\right]\Rightarrow \left[\begin{array}{ccc}-1& 0& 5\\ 0& 1& 1\\ -1& 0& 4\end{array}\right]=\left[\begin{array}{ccc}3& 2& 0\\ 1& 1& 0\\ 0& 2& 1\end{array}\right]$

$\left[\begin{array}{ccc}-1& 0& 5\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}3& 2& 0\\ 1& 1& 0\\ 3& 0& -1\end{array}\right]\Rightarrow \left[\begin{array}{ccc}+1& 0& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}-3& 0& 1\\ 1& 1& 0\\ 0& 2& 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}-3& 0& 1\\ 1& -1& -1\\ 0& 2& 1\end{array}\right]=A^{-1}$.