# Product of two matrices with zero product

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.

• Like you do with numbers, multiply, from the left, both sides by $A^{-1}$. – orole Jan 18 '18 at 20:56

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 $A$""""".
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?
• A $n \times m$ matrix defines a linear transformation $T_A\colon \Bbb R^m \to \Bbb R^n$, right? The nullity of $A$ is defined as the nullity of $T_A$. – Ivo Terek Jan 18 '18 at 21:07
If $A$ is non-singular, $A^{-1}$ exists.
Premultiply $A^{-1}$ to the equation to obtain the desired result.
If $A$ is nonsingular then it is invertible. Thus, B = $A^{-1}\cdot 0 = 0$.