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Math Help - Simple Matrix Proof

  1. #1
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    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.
    Last edited by JDUBC; September 15th 2011 at 12:19 PM.
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  2. #2
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    Re: Simple Matrix Proof

    Quote Originally Posted by JDUBC View Post
    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~?
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  3. #3
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    Re: Simple Matrix Proof

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