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

    Thanks
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  2. #2
    MHF Contributor

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    The only way for the the product AB to exist is for A to be a j \times m and for B to be a m \times n.
    If the product is square what does that tell us about j~\&~n?
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  3. #3
    Senior Member Danneedshelp's Avatar
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    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
    j=n=m.
    It does mean tha j=n. But not necessarily j=n\color{red}=m
    What about a m \times n times a j \times m?
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