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Math Help - finding a matrix given the inverse

  1. #1
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    finding a matrix given the inverse

    how would i find a matrix with inverse |1 4|
    |2 5|
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  2. #2
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    Quote Originally Posted by cottekr View Post
    how would i find a matrix with inverse |1 4|
    |2 5|
    Matrix Inverse -- from Wolfram MathWorld

    The 2x2 matrix has an explicit formula for calculating its inverse. Put a,b,c and d back in their original location and then factor out a \frac{1}{ad-bc} from every term.
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  3. #3
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    Jameson gave you references for finding the inverse of a matrix. You are asking "given the inverse, how do I find the matrix" or "Given A^{-1} how do I find A?".

    To do that, you need to know that the inverse is "dual"- doing it again puts you back where you started: A= (A^{-1})^{-1}.

    In other words, just find the inverse of \begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix} using the methods Jameson is referring to.

    Since this is a simple 2 by 2 problem, you could also do it, right from the definition, like this:
    Write the original matrix as A= \begin{bmatrix}a & b \\ c & d\end{bmatrix}. The definition of "inverse matrix" says that AA^{-1}= I so we must have

    AA^{-1}= \begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 & 4 \\ 2 & 5\end{bmatrix}= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}

    so
    \begin{bmatrix}a+ 2b & 4a+ 5b \\ c+ 2d & 4c+ 5d\end{bmatrix}= \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}.

    That gives you the four equations a+ 2b= 1, 4a+ 5b= 0, c+ 2d= 0, and 4c+ 5d= 1, to solve for a, b, c, and d.
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