For a singular matrix A, the determinant is zero: det(A)=0.
There is a fundamental property relating a matrix to its adjoint:
A(adj A)=det(A)×I
where I is the identity matrix.
Since A is singular, det(A)=0.
Substituting into the formula:
A(adj A)=det(A)×I
A(adj A)=0×I
A(adj A)=O
where O is the null matrix (a matrix with all elements equal to zero).
Therefore, A(adj A) is a null matrix.