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Math Help - Matrix equation

  1. #1
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    Matrix equation

    Given the matrix equation

    X^{-1}A^{T} + B^{-1}C = 0

    Where A,B,C, and X are 2 x 2 invertible matrices. Solve for X in terms of A,B,C.

    ii) Find X when A = \begin{matrix} [1 & 1] \\ [0 & 2]  \end{matrix}, B= \begin{matrix} [1 & 2] \\ [1 & 1] \end{matrix}, C= \begin{matrix} [1 & 2] \\ [1 & 4] \end{matrix}
    Last edited by dumbledore; October 19th 2012 at 07:39 AM.
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  2. #2
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    Lightbulb Re: Matrix equation

    x=\left(\begin{array}{cc} 1 & 3 \\ 1 & 2\end{array}\right)
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  3. #3
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    Re: Matrix equation

    For solving for X (part i) of the question , is this correct?

    B^{-1} C = - X^{-1} A^{T}

    left multiply by X

    X B^{-1} C = - (X X^{-1}) A^{T}

    X B^{-1} C = - I A^{T}

    right multiply by C^{-1} B

    X B^{-1} C (C^{-1} B) = - A^{T} (C^{-1} B)

    X = - A^{T} C^{-1} B

    as

    B^{-1} C (C^{-1} B) = B^{-1} (C C^{-1}) B =

    = B^{-1} I B = B^{-1} B = I
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  4. #4
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    Re: Matrix equation

    Yes, of course that is correct.
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  5. #5
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    Re: Matrix equation

    Thanks HallsofIvy. The problem now is that when I plug in my 2x2 matrices to the equation X = -A^{T} C^{-1} B , I do not get the matrix that MaxJasper provided above.
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  6. #6
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Matrix equation

    Typo: Sorry, minus sign was dropped, so correct x is:

    \text{x}=-\left(\begin{array}{cc} 1 & 3 \\ 1 & 2\end{array}\right)
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  7. #7
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    Re: Matrix equation

    Got it!
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