I've been asked to prove that the following statements are equivalent for any $n$x$n$ matrix $A$: (Note that A and B both are $n$x$n$)
- $AB = 0$ for some non-zero $B$
- $CA = 0$ for some non-zero $C$
How do I show this equivalence? I'm able to see that if $(1)$ holds, the columns of A are linearly dependent and the rows of B are linearly dependent. How do I go ahead?
P.S. The course I'm currently doing, however, has only covered concepts of matrix multiplication, elementary matrices, and system of equations - so it'd be great if you could provide a proof along those lines!
Also, I was wondering if there's a general closed-form or some way we can describe matrices A that satisfy $(1)$ and $(2)$?