Question

Let A and B be nxn matrices such that AB is singular prove that either A or Bis singular

Answer 1

$\det(\mathbf{AB})=\det\mathbf A\times\det\mathbf B$

Because $\mathbf{AB}$ is singular, we have

$\det(\mathbf{AB})=0$

from which it follows that either $\det\mathbf A=0$ or $\det\mathbf B=0$.