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Math Help - [SOLVED] 2 Matrix questions, Invertible and A^2=A

  1. #1
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    [SOLVED] 2 Matrix questions, Invertible and A^2=A

    I have no idea what to do for either of these questions

    1. If A = |x -1|
    |y -1|

    determine the value of x and y such that A^2=A



    2. The matrix |2 -9|
    |9 k|

    is invertible if and only if k !=??
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  2. #2
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    1. <br /> <br />
A^2=A \rightarrow A^2-A=(A-I_2)A=0 <br />
\rightarrow im(A)\in \ker(A-I_2) <br />

    <br />
 \rightarrow im(A)=span(\begin{bmatrix}x\\y\end{bmatrix},<br />
 \begin{bmatrix}-1 \\-1 \end{bmatrix})<br />
 \rightarrow <br />
 \left.\begin{array}{cc}<br />
 (A-I_2)\begin{bmatrix}x\\y\end{bmatrix}=0 \\<br />
 (A-I_2)\begin{bmatrix}-1 \\-1 \end{bmatrix}=0 \\<br />
 \end{array}\right\} \\<br />
 \rightarrow x=y=2<br /> <br />

    2. \det\begin{bmatrix}2&-9\\9&k\end{bmatrix}=<br />
2k+81\neq 0 \rightarrow k\neq -\frac{81}{2}<br />
    Last edited by math2009; February 9th 2010 at 04:22 AM.
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  3. #3
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    thanks
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