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Math Help - Matrix Inverse Problem

  1. #1
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    Joined
    Oct 2010
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    8

    Matrix Inverse Problem

    Problem:
    Solve the matrix equation for X
    \begin{pmatrix}<br />
3 & 2 \\<br />
1 & 0<br />
\end{pmatrix}<br />
X_(2,3) <br />
\begin{pmatrix}<br />
1 & -1 & 1 \\<br />
2 & 1 & 0 \\<br />
0 & 0 & 2<br />
\end{pmatrix}<br />
=<br />
\begin{pmatrix}<br />
1 & 2 & 1 \\<br />
4 & -1 & 1<br />
\end{pmatrix}

    Attempt at a Solution:
    I started working on this problem thinking I could take the inverse of both the matrices being multiplied by X_2,3 and then multiply each one by the matrix on the right hand side. Well that worked for the 2,2 matrix, but then I remembered you can't multiply a 3,3 matrix and a 2,3 matrix. I know I could transpose the 2,3 matrix to a 3,2 matrix so I could multiply it by the 3,3 matrix, but how would this affect my answer? Also is this the way to approach this problem or is there an easier way? All advice/help is appreciated!
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  2. #2
    Master Of Puppets
    pickslides's Avatar
    Joined
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    Melbourne
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    Assuming you mean X is to be a 2x3 matrix then follow this

    AXB=C

    A^{-1}AXB=A^{-1}C

    XBB^{-1}=A^{-1}CB^{-1}

    X=A^{-1}CB^{-1}

    Therefore


    X_{(2,3)} <br /> <br />
=<br /> <br />
\begin{pmatrix}<br />
3 & 2 \\<br />
1 & 0<br />
\end{pmatrix}^{-1}<br />
\begin{pmatrix}<br />
1 & 2 & 1 \\<br />
4 & -1 & 1<br />
\end{pmatrix} <br /> <br />
\begin{pmatrix}<br />
1 & -1 & 1 \\<br />
2 & 1 & 0 \\<br />
0 & 0 & 2<br />
\end{pmatrix}^{-1}
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  3. #3
    Newbie
    Joined
    Oct 2010
    Posts
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    Oh okay, so that way I don't have to transpose the 2,3 matrix. Awesome! Thanks very much!
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