Answer
$$\operatorname{adj}(A)=\left[ \begin {array}{ccc} 0&0&0\\ 0&-12&-6
\\ 0&4&2\end {array} \right].
$$
$A$ is singular matrix and has no inverse.
Work Step by Step
The matrix is given by $A=\left[ \begin {array}{ccc} 1&0&0\\ 0&2&6
\\ 0&-4&-12\end {array} \right]
.$ To find $\operatorname{adj}(A)$, we calculate first the cofactor matrix of $A$ as follows
$$\left[ \begin {array}{ccc} 0&0&0\\ 0&-12&4
\\ 0&-6&2\end {array} \right]
.$$
Now, the adjoint of $A$ is
$$\operatorname{adj}(A)=\left[ \begin {array}{ccc} 0&0&0\\ 0&-12&-6
\\ 0&4&2\end {array} \right]
$$
To find $A^{-1}$, we have to calculate $\det(A)$ which is given by
$$\det(A)=0.$$
Hence, $A$ is singular matrix and has no inverse.
