Describe All the solutions to: AX=0 (Square Matrices, A≠0)

1. Oct 15, 2012

Swimmingly!

1. The problem statement, all variables and given/known data
A and X are square matrices. A≠0

Describe all solutions to:
AX=0

2. Relevant equations
3. The attempt at a solution
X$_{i}$ is some solution.
Solutions:
• X=0
• k*X$_{i}$
• ƩX$_{i}$

Looking for other solutions:
Let there be B: AB=BA
AX=0 ⇔ BAX=B0 ⇔ ABX=0

New solution: BX

So maybe all solutions are describable as:
BX=0, AB=BA

But this needs proof! (and may allow for further refining of the solution)

Other info:
If there exists X$_{i}$≠0 => A$^{-1}$ doesn't exist. (X=A$^{-1}$0=0 False)

BX$_{1}$=X$_{2}$ ⇔ X$_{1}$=B$^{-1}$X$_{2}$
What if B has no inverse? Then you've gotta state your solution with X$_{2}$. But this may keep up, so is there any mother matrix from which these sprout? I think I found B without inverse.