Question

Find the adjoint of the matrix $A=\left(\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right)$ and
hence show that

$A.\left(adjA\right)=|A|I_3$.

Answer 1

$|A|=\left|\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right|=27$

Adjoint of a matrix is the transpose of its' cofactor matrix.

$adj\left(A\right)=\left[\begin{array}{ccc}-3& 6& 6\\ -6& 3& -6\\ -6& -6& 3\end{array}\right]$

$A.adj\left(A\right)=\left[\begin{array}{ccc}-1& -2& -2\\ 2& 1& -2\\ 2& -2& 1\end{array}\right]\times \left[\begin{array}{ccc}-3& 6& 6\\ -6& 3& -6\\ -6& -6& 3\end{array}\right]=\left[\begin{array}{ccc}27& 0& 0\\ 0& 27& 0\\ 0& 0& 27\end{array}\right]=27\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Hence proved.