Matrix without an inverse

Question

Using the definition of an inverse, can someone explain why $0_n$$_x$$_n$ cannot have an inverse.

Also can someone explain if AB=$0_n$$_x$$_n$ for two nxn nonzero matrices A and B, then how A nor B can have an inverse.

Answer 1

$A$ has an inverse iff there is $B$ where $AB=I.$ If $A=0,$ then no matter what $B$ is we get $AB=0,$ which is not $I.$ I leave the other question to you...

Answer 2

Suppose that $A$ and $B$ ae invertble. Then $0=A^{-1}ABB^{-1}=I$. Absurd !