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Math Help - Gauss-Jordan elimination

  1. #1
    Super Member craig's Avatar
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    Gauss-Jordan elimination

    Normally have no problem with these types of problems to find the inverse of a matrix. However I'm not sure how to start with this one, usually the top left value is rather conveniently 1.

    Find A^{-1} where A = \left( \begin{array}{ccc} -1 & -2 & 2 \\ -2 & -1 & 2 \\ -2 & -2 & 3 \\ \end{array} \right).

    Do I multiply A by -1 and then proceed as normal?

    Thanks for your help
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  2. #2
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    Not sure which method you have been using for finding A^{-1}, but if you think about the augmented 3x6 matrix with the last 3 columns as the Identity matrix (3x3 matrix with 1's down the diagonal), then doing GJ elimination on this matrix should yield a 3x6 matrix with the first 3 columns being the identity matrix and the last 3 columns now being A^{-1}. So to change {-1,-2,2} to {1,2,-2} you would multiply the whole first row {-1,-2,2,1,0,0} by -1 and get {1,2,-2,-1,0,0}. You can proceed through rows 2 and 3 the same way.
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  3. #3
    Super Member craig's Avatar
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    Quote Originally Posted by Bgrasty View Post
    Not sure which method you have been using for finding A^{-1}, but if you think about the augmented 3x6 matrix with the last 3 columns as the Identity matrix (3x3 matrix with 1's down the diagonal), then doing GJ elimination on this matrix should yield a 3x6 matrix with the first 3 columns being the identity matrix and the last 3 columns now being A^{-1}. So to change {-1,-2,2} to {1,2,-2} you would multiply the whole first row {-1,-2,2,1,0,0} by -1 and get {1,2,-2,-1,0,0}. You can proceed through rows 2 and 3 the same way.
    Thanks for the reply.

    So I'd just multiply the first row by -1, and do nothing to the second and third ones?
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  4. #4
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    After you multiply the first row by -1, add 2 times the first row to the second row and 2 times the first row to the 3rd row. This leaves the first row as the only one with a number in the first column.
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  5. #5
    Super Member craig's Avatar
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    Quote Originally Posted by kaylakutie View Post
    After you multiply the first row by -1, add 2 times the first row to the second row and 2 times the first row to the 3rd row. This leaves the first row as the only one with a number in the first column.
    Thankyou.
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