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Math Help - Help getting started on a problem (intro linear algebra)

  1. #1
    Senior Member Danneedshelp's Avatar
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    Help getting started on a problem (intro linear algebra)

    Q: Suppose A=\begin{bmatrix}a&b\\c&d&\end{bmatrix}. Find conditions on a,b,c,and d for which A^{-1}=A. Then find two different matrices, other than the identity matrix, for which A^{-1}=A.

    Do I just set up an equation to find what A^{-1} is and then equate that to A to find out when A^{-1}=A?

    Or could I use the face that if A^{-1}=A \Rightarrow A^{2}=I?

    Im not really sure to I pull everything together.

    Thanks


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  2. #2
    Eater of Worlds
    galactus's Avatar
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    You are correct. If A=A^{-1}, then A^{2}=I

    \begin{bmatrix}a&b\\c&d\end{bmatrix} \cdot \begin{bmatrix}a&b\\c&d\end{bmatrix}=\begin{bmatri  x}1&0\\0&1\end{bmatrix}

    \begin{bmatrix}a^{2}+bc&ab+bd\\ac+cd&bc+d^{2}\end{  bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}

    a^{2}+bc=1
    ab+bd=0
    ac+cd=0
    bc+d^{2}=1

    One such matrix that fits is \begin{bmatrix}1&0\\-3&-1\end{bmatrix}
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