Product of two matrices with zero product

Question

If there are two matrices, lets say A & B such that $$ AB = 0 $$ and A is a non singular matrix and B may or may not be a square matrix . Can we infer anything about nature of B . The book says B is a zero matrix but I am unable to prove.

Answer 1

If $A$ is non-singular, there exists $A^{-1}$. So $$AB = 0 \implies A^{-1}(AB) = A^{-1}0 \implies (A^{-1}A)B = 0 \implies {\rm Id}\; B = 0 \implies B = 0.$$The moral of the history is that if $A$ is non-singular, you can """""divide by $A$""""".

Answer 2

Here's another approach: If the nullity of $AB$ is $n$, the nullity of $A$ plus the nullity of $B$ must be at least $n$. Can you see why?

Answer 3

If $A$ is non-singular, $A^{-1}$ exists.

Premultiply $A^{-1}$ to the equation to obtain the desired result.

Answer 4

If $A$ is nonsingular then it is invertible. Thus, B = $A^{-1}\cdot 0 = 0$.