A doubt on Matrix Multiplication

Question

Suppose I have two matrices $A$ and $B$ satisfying the equation $A B = O$, where $O$ is a null matrix. I want to claim something about $A$ and $B$ from this equation. I know that we cannot claim that neither of $A$ and $B$ may be zero. What can I claim? Can I claim something about the determinant values of $A$ and $B$ ? Please help.

Answer 1

You know that $\det(A \cdot B) = \det(O) = 0$, and that the determinant is a multiplicative function, that is, $\det(A \cdot B) = \det A \cdot \det B$.

Then, you have that, if your matrices are with values on a domain (for example, $\mathbb{R}$ or $\mathbb{C}$), the two information implies $\det(A) = 0$ or $\det(B) = 0$.

Answer 2

If $ AB= O_n $ then $ \det(AB) = \det(O_n) = 0 \iff \det(A) \det(B) = 0$ which means that either $\det(A) = 0 $ or $\det(B) = 0$