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Thread: Simple matrix algebra identity

  1. #1
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    Simple matrix algebra identity

    Hello everyone!

    I'm stuck on this particular concept...
    $\displaystyle AB=AC \ \Rightarrow \ A(B-C)=0.$ Soooo when is it that $\displaystyle B-C=0$ i.e. $\displaystyle B=C \ ... \ (1)$ but probably if $\displaystyle A^{-1}$ exists then we can multiply to the right by $\displaystyle A^{-1}$ so we get $\displaystyle (1)$. But how can we know before hand?
    Thanks!
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  2. #2
    A Plied Mathematician
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    If you know that $\displaystyle \det(A)\not=0$, then $\displaystyle A$ is invertible, which would imply that $\displaystyle B=C.$ That's one condition that is sufficient. It's not necessary. As a counterexample, just let $\displaystyle A=0$, and $\displaystyle B=C=I.$

    Note: you're actually going to want to multiply on the left by $\displaystyle A^{-1}$, if it exists, in order to eliminate the $\displaystyle A$'s.
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