Question

Find the inverse of matrix by adjoint method
$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 3& 1& 2\end{array}\right]$

Answer 1

$A=\left(\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 3& 1& 2\end{array}\right)$
$|A|=1(6-1)-2(4-3)+3(2-6)$
$5-2-12$
$-9$
Minor of A$=\left(\begin{array}{ccc}5& 1& -4\\ 1& -4& -5\\ -4& -5& -1\end{array}\right)$
$Cof\left(A\right)=\left(\begin{array}{ccc}5& -1& -4\\ -1& -4& +5\\ -4& +5& -1\end{array}\right)$
adj (A)$=\left(\begin{array}{ccc}5& -1& -4\\ -1& -1& 5\\ -4& 5& -1\end{array}\right)$
$A^{-1}=\frac{adj\left(A\right)}{|A|}=\frac{1}{9}\left(\begin{array}{ccc}-5& 1& 4\\ 1& 4& -5\\ 4& -5& 1\end{array}\right)$
$=\frac{1}{9}\left(\begin{array}{ccc}-5& 1& 4\\ -1& 4& -5\\ 4& -5& 1\end{array}\right)$