If $A$ is a singular matrix, then $\text{adj }A$ is:

By the property of the determinant of the adjoint of a matrix:
$$A\cdot \text{adj }A=|A|I$$
Taking the determinant on both sides:
$$|A\cdot \text{adj }A|=||A|I|$$
$$|A|\cdot |\text{adj }A|=|A{|}^n$$
$$|\text{adj }A|=|A{|}^{n-1}$$
where $n$ is the order of matrix $A$.Given that $A$ is a singular matrix, we have $|A|=0$.Substituting this into the formula:
$$|\text{adj }A|=0^{n-1}=0$$
Since the determinant of $\text{adj }A$ is $0$, $\text{adj }A$ is also a singular matrix.