Question

Find the inverse of the matrix (if it exists)

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

Answer 1

Inverse of $A$ is is $\left(A^{-1}\right)=\frac{adjA}{|A|}$
Now, adj $A$ is calculated in two steps
1) find cofactors of given matrix $A$
2) then take transpose of cofactor matrix
$A=\left|\begin{array}{ccc}1& 0& 0\\ 3& 3& 0\\ 5& 2& -1\end{array}\right|$
Cofactor of $A$ (say $B$) is
$B=\left|\begin{array}{ccc}-3& 3& -9\\ 0& -1& -3\\ 0& 0& 3\end{array}\right|$
$adjA=B^T=\left|\begin{array}{ccc}-3& 0& 0\\ 3& -1& 0\\ -9& -3& 3\end{array}\right|$
$Det\left(A\right)=-3$
$A^{-1}=\frac{adjA}{|A|}=\frac{1}{-3}\left|\begin{array}{ccc}-3& 0& 0\\ 3& -1& 0\\ -9& -3& 3\end{array}\right|$