Answer
$$\operatorname{adj}(A)=\left[ \begin {array}{ccc} -1&1&1\\ -1&1&1
\\ 1&-1&-1\end {array} \right]
.$$
$A$ is singular and has no inverse.
Work Step by Step
The matrix is given by $A=\left[ \begin {array}{ccc} 0&1&1\\ 1&2&3
\\ -1&-1&-2\end {array} \right]
.$
To find $\operatorname{adj}(A)$, we calculate first the cofactor matrix of $A$ as follows
$$\left[ \begin {array}{ccc} -1&-1&1\\ 1&1&-1
\\ 1&1&-1\end {array} \right]
.$$
Now, the adjoint of $A$ is
$$\operatorname{adj}(A)=\left[ \begin {array}{ccc} -1&1&1\\ -1&1&1
\\ 1&-1&-1\end {array} \right]
.$$
To find $A^{-1}$, we have to calculate $\det(A)$ which is given by
$$\det(A)=0.$$
So, $A$ is singular and has no inverse.