Matrix proof on the zero matrix

Question

I have looked online and can't seem to find a solution, but I am struggling with this question and was wondering if anyone would be able to help me.

Prove that for two $2\times 2$ matrices $A$ and $B$, where neither $A$ nor $B$ is the zero matrix $Z$, that if $AB=Z$ then both $A$ and $B$ must be singular.

Any help would be greatly appreciated.

Answer 1

For sake of contradiction, suppose that $AB =Z$, $A \neq Z$, $B \neq Z$, AND $A$ is not singular. Then $A$ is invertible with inverse $A^{-1}$. Then from $AB=Z$, premultiplying on both sides by $A^{-1}$ yields $B = A^{-1}Z$. But $A^{-1}Z = Z$, contradicting our assumption that $B$ is non-zero. Similarly, assuming $B$ is not singular yields that $A = Z$. Ergo, both $A$ and $B$ must be singular.

Answer 2

Different approach using contradiction: use the fact that $\det (AB) = \det A \det B $ and that $A$ singular $\Rightarrow \det A = 0$.