Chapter 1.7, Problem 52E

If $A$ and $B$ are $\left(n\times n\right)$ matrices such that $A$ is nonsingular and $AB=\mathcal{O}$, then prove that $B=\mathcal{O}$. [Hint: Write $B=\left[{\mathbf{B}}_1,\dots ,{\mathbf{B}}_n\right]$ and consider $AB=\left[A{\mathbf{B}}_1,\dots ,{\mathbf{A}\mathbf{B}}_n\right]$.]