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Thread: Simple Matrix Proof

  1. #1
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    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.
    Last edited by JDUBC; Sep 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 $\displaystyle B_{mxm}$ 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 $\displaystyle A~?$
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  3. #3
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    Re: Simple Matrix Proof

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