Given an $n\times n$ square matrix $A$, if $C(A) ⊆ N(A)$, then $A^2$ is the $n\times n$ zero matrix. Why is this the case?

Question

Given a $n\times n$ square matrix $A$, if $C(A) ⊆ N(A)$ - where $C$ and $N$ are respectively the column space and the null space - then $A^2$ is the $n\times n$ zero matrix. Why is this the case?

I would be grateful for a concise explanation of the theorem in the question.

Answer 1

Hint If $x \in \mathbb R^n$ then $Ax \in C(A) \subseteq N(A)$ and hence $$A(Ax)=0$$