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

[Swimmingly!]

Homework Statement

A and X are square matrices. A≠0

Describe all solutions to:
AX=0

Homework Equations

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 got to 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.

 

[Asim Riaz]

Conclusion:All solutions to AX=0 are describable as:X=0, k*X_{i}, ƩX_{i}, BX, BX_{1}=X_{2} (B has inverse)