# Thread: Simple Matrix Proof

1. ## Simple Matrix

Hey guys quick simple question I'm having some troubles with,

Define $B_{mxm}$ and $A_{mxn}$ where $A$ is non-zero. Suppose $AB=0$

Show that no matrix $C_{mxm}$ exists such that $CB=I_m$

Trivially if B is a 0 matrix then the above is self-evident. it's in situations when it is not 0 that i am having trouble. Bellow is an example where this is true,

$A= \begin{bmatrix}\ 2& 3 \\ 0 & 0\end{bmatrix}$ and $B=\begin{bmatrix}\ 3& 3 \\ -2 & -2\end{bmatrix}$

$AB=\begin{bmatrix}\ 0& 0 \\ 0 & 0\end{bmatrix}$

But since there is no inverse of B we cannot have $CB=I_m$

So I have one case where this is true but I can't see the connection in a general sense. I suppose we need to show that any matrix B that satisfies the above doesnt have an inverse.

2. ## Re: Simple Matrix Proof

Originally Posted by JDUBC
Hey guys quick simple question I'm having some troubles with,
Define $B_{mxm}$ and $A_{mxn}$ where $A$ is non-zero. Suppose $AB=0$
Show that no matrix $C_{mxm}$ exists such that $CB=I_m$
What if $B^{-1}$ does exist?
What does that say about $A~?$

3. ## Re: Simple Matrix Proof

Originally Posted by Plato
What if $B^{-1}$ does exist?
What does that say about $A~?$
Thanks very much!