Question

If A and B are two non-zero square matrices of the same order then AB = O implies that both A and B must be singular.

Answer 1

Let us assume that A is non-singular i.e. |A| $\ne$ 0 and hence $A^{-1}$ exists such that $A{A}^{-1}=I$.
$\therefore AB=0$
$⟹A^{-1}\left(AB\right)=\left(A^{-1}A\right)B=IB=B=0$
Above shows that B is a null matrix which is a contradiction.
Similarly, if B is non-singular then as above we will have A=0 which is again a contradiction. hence, both A and B must be singular.