Prove that if A is singular, then adj(A) is also singular Prove that if A is singular, then adj(A) is also singular.
How do you prove this without proving by contradiction?

From the basic equation: $A\,\text{adj}(A)=\det(A)I$ we have:
$$
\det(A\,\text{adj}A)=\det(\det(A)I)\\
\det(A)\det(\text{adj}(A))=\det(A)^n \det(I)
$$
$\det(\text{adj}(A))=\det(A)^{n-1}$
Since $A$ is singular,
$$
\det(\text{adj}(A))=0 
$$
and then $\text{adj}(A)$ is also singular 
