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

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

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

    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 $\displaystyle A=A^{-1}$, then $\displaystyle A^{2}=I$

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

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

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

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