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Thread: ERO's and ECO's

  1. #1
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    ERO's and ECO's

    Let A be an $\displaystyle m \times n$ matrix with entries in $\displaystyle \mathbb{R}$. Show by using elementary row and column operations, that there are invertible matrices P and Q (where P has size $\displaystyle m \times m$ and Q has size $\displaystyle n \times n$) such that:
    $\displaystyle PAQ=\[\left( \begin{array}{cc} I_r & 0 \\ 0 & 0 \\ \end{array} \right)\] $.

    I can sort of see why this is true. You would need to use ECO's to make all but 1 entries in a row 0, and then permute the rows until you get the requied order. But I can't see how I can present this in the form of a solution. Help anyone?
    Last edited by worc3247; Dec 30th 2010 at 09:29 AM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by worc3247 View Post
    Let A be an mxn matrix with entries in $\displaystyle \mathbb{R}$. Show by using elementary row and column operations, that there are invertible matrices P and Q (where P has size $\displaystyle m \cross m$ and Q has size $\displaystyle n \cross n$) such that:
    $\displaystyle PAQ=\[\left( \begin{array}{cc} I_r & 0 \\ 0 & 0 \\ \end{array} \right)\] $.

    I can sort of see why this is true. You would need to use ECO's to make all but 1 entries in a row 0, and then permute the rows until you get the requied order. But I can't see how I can present this in the form of a solution. Help anyone?

    This isn't true. Take $\displaystyle m=n$ then the above would imply that $\displaystyle \displaystyle \det(A)=\frac{1}{\det(PQ)}\det\begin{pmatrix}I_r & 0\\ 0 &0\end{pmatrix}=0$ and so this isn't true if $\displaystyle A\in\text{GL}_n\left(\mathbb{R}\right)$. Right?
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    I suppose worc3247 meant "... and $\displaystyle \textrm{rank}(A)=r$ " then, the statement is true.

    Fernando Revilla
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    I can see what you mean. Though I did write out the question excactly as it is on my problem sheet.
    Nevertheless if someone could help me to explain why this is correct (assuming rank(A)=r) I would appreciate it.
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by worc3247 View Post
    I can see what you mean. Though I did write out the question excactly as it is on my problem sheet.
    Nevertheless if someone could help me to explain why this is correct (assuming rank(A)=r) I would appreciate it.
    Perhaps this can help you.

    Fernando Revilla
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