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Math Help - singular matrices, transposes etc

  1. #1
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    singular matrices, transposes etc

    is the transpose of a singular Matrix singular?

    and what about if A is singular, is A^T . A singular?

    also


    I know if you augment a non-singular square matrix with the identity matrix of the same dimension the you will get (inverse|identity) if you reduce it to echelon form

    but what about non-square matrices

    what does row reducing (matrix|identity) do??? and what about then doing (transpose(matrix)|identity)??

    manny thanks
    Last edited by James0502; February 17th 2009 at 03:57 PM.
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  2. #2
    MHF Contributor arbolis's Avatar
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    is the transpose of a singular Matrix singular?
    The answer is yes. A matrix A is singular if and only if its determinant is equal to 0.
    There's a proof that says that \det A= \det A^T. So if A is singular, so is its transpose. And if a A^T is singular, so is A.


    And about
    but what about non-square matrices

    what does row reducing (matrix|identity) do??
    You cannot form "identity" since you don't have a square matrix.
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  3. #3
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    but what about if you have an n x m matrix and augment it with an m x m identity, then row reduce? what happens?

    (thanks)
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  4. #4
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by James0502 View Post
    but what about if you have an n x m matrix and augment it with an m x m identity, then row reduce? what happens?

    (thanks)
    Nothing would happen. If you do that and say that m>n, then you wouldn't be able to touch the lowest rows of the identity matrix.
    You can only do that when you have a square matrix. And this method finds the inverse of the matrix you chose, if it has an inverse. If you chose a non square matrix it's not even worth to bother trying to reduce it with an mxm identity matrix aside it. As I said, you wouldn't find anything.
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