1. ## Simple Matrix

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

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

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

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,

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

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

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

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 $\displaystyle B_{mxm}$ and $\displaystyle A_{mxn}$ where $\displaystyle A$ is non-zero. Suppose $\displaystyle AB=0$
Show that no matrix $\displaystyle C_{mxm}$ exists such that $\displaystyle CB=I_m$
What if $\displaystyle B^{-1}$ does exist?
What does that say about $\displaystyle A~?$

3. ## Re: Simple Matrix Proof

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