The column space of $A^2$ is all of $\mathbb R^n$ if and only if the column space of $A$ is all of $\mathbb R^n$

Question

How would I go about proving the following statement?

"Let $A$ be an $n \times n$ matrix. $\operatorname{Col}(A^2)=\mathbb{R}^n$ if and only if $\operatorname{Col}(A)=\mathbb{R}^n$"

I started off by proving that $A$ is linearly independent if and only if $Ax=0$ which implies that $x=0$. I showed that the columns of A span $\mathbb{R}^n$ if and only if $Ax=b$ has a solution for every b in $\mathbb{R}^n$ so that for any b, $x= A^{-1}b$.

I'm confused about how to use this to prove that $\operatorname{Col}(A^2)$ is in the span $\mathbb{R}^n$

Answer 1

For any $b$ you can find $x$ such that $Ax = b$. Now given such an $x$, you can find $y$ such that $Ay = x$. But then $A^2y = b$.