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Thread: Simple proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    Simple proof

    Hello, I just wanted to make sure my proof is correct before turning it in. It's a very basic proof, but it's been a while since I have proved anything.

    Q: Prove that if the product AB is a square matrix, then the product BA is defined.

    A: Suppose the product $\displaystyle AB$ is not a square matrix. Then, by the definition of matrix multiplication, $\displaystyle A$ must be an $\displaystyle m\times$n matrix and $\displaystyle B$ an $\displaystyle n\times$p matrix where $\displaystyle m\neq{p}$(by out supposition). It follows that $\displaystyle BA$is not defined, because the column $\displaystyle p$ of matrix $\displaystyle B$ is not equal to row $\displaystyle m$ of matrix $\displaystyle A$. But, our original statement reads $\displaystyle BA $is defined. Therefore, by contradiction we have proven that if the product $\displaystyle AB$is a square matrix, then the product $\displaystyle BA$ is defined.

    Thanks
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  2. #2
    MHF Contributor

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    The only way for the the product $\displaystyle AB$ to exist is for $\displaystyle A$ to be a $\displaystyle j \times m$ and for $\displaystyle B$ to be a $\displaystyle m \times n$.
    If the product is square what does that tell us about $\displaystyle j~\&~n$?
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  3. #3
    Senior Member Danneedshelp's Avatar
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    $\displaystyle j=n=m$

    ...right?

    So, I should just go with a direct proof.
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  4. #4
    MHF Contributor

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    Quote Originally Posted by Danneedshelp View Post
    $\displaystyle j=n=m$.
    It does mean tha $\displaystyle j=n$. But not necessarily $\displaystyle j=n\color{red}=m$
    What about a $\displaystyle m \times n$ times a $\displaystyle j \times m$?
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