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Math Help - Finding an inverse matrix using Gauss-Jordan

  1. #1
    m58
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    Finding an inverse matrix using Gauss-Jordan

    Hello,

    I've been trying to find the inverse of the following matrix for some time now:

    1 0 2
    2 -1 3
    4 1 8

    I can easily find the answer on my calculator, but I have no idea how to find the inverse! I performed a basic Gauss-Jordan, and got an identity matrix as a result; the answer is really:
    -11 2 2
    -4 0 1
    6 -1 -1

    Thanks in advance!
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  2. #2
    MHF Contributor
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    Re: Finding an inverse matrix using Gauss-Jordan

    Did you add the augmented Identity Matrix BEFORE you started the Gauss-Jordan process? The fact that you showed only a 3x3 is not encouraging.

    Start with 3x3.
    Augment (add to the right) with Identity. It is now 3x6
    Do Gauss-Jordan. Left 3 are identity. Right 3 are Inverse.
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  3. #3
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    Re: Finding an inverse matrix using Gauss-Jordan

    Set up your matrix like this, like TKHunny said.

    \left[\begin{array}{ccc|ccc}1 & 0 & 2 & 1 & 0 & 0 \\ 2 & -1 & 3 & 0 & 1 & 0\\ 4 & 1 & 8 & 0 & 0 & 1\end{array}\right]

    Now row-reduce as usual; your goal is to get the 3x3 identity matrix on the left. Once you've done that, the 3x3 matrix on the right will be the inverse.

    To get you started, the first thing I would do is add -2 times the 1st row to the 2nd row:

    \left[\begin{array}{ccc|ccc}1 & 0 & 2 & 1 & 0 &  0 \\ 0 & -1 & -1 & -2 & 1 & 0\\ 4 & 1 & 8  & 0 & 0 & 1\end{array}\right]

    After several steps of row-reducing, you'll get the the form:

    \left[\begin{array}{ccc|ccc}1 & 0 & 0 & -11 & 2 & 2 \\ 0 & 1 & 0 & -4 & 0 & 1\\ 0 & 0 & 1  & 6 & -1 & -1\end{array}\right]

    As you can see, the left 3x3 of the matrix is now the identity matrix and the right half is your inverse that you noted in the original post.
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  4. #4
    m58
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    Re: Finding an inverse matrix using Gauss-Jordan

    Ah, OK! I think I've got it now...


    Add (-2 * row1) to row2
    1 0 2 1 0 0
    0 -1 -1 -2 1 0
    4 1 8 0 0 1

    Add (-4 * row1) to row3
    1 0 2 1 0 0
    0 -1 -1 -2 1 0
    0 1 0 -4 0 1

    Divide row2 by -1
    1 0 2 1 0 0
    0 1 1 2 -1 0
    0 1 0 -4 0 1

    Add (-1 * row2) to row3
    1 0 2 1 0 0
    0 1 1 2 -1 0
    0 0 -1 -6 1 1

    Divide row3 by -1
    1 0 2 1 0 0
    0 1 1 2 -1 0
    0 0 1 6 -1 -1

    Add (-1 * row3) to row2
    1 0 2 1 0 0
    0 1 0 -4 0 1
    0 0 1 6 -1 -1

    Add (-2 * row3) to row1
    1 0 0 -11 2 2
    0 1 0 -4 0 1
    0 0 1 6 -1 -1

    (I do each row one by one, to avoid errors)

    Thanks for your help!
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