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Find a 3x3 nondiagonal matrix whose eigenvalues are $-2,-2,$ and $3$, and associated eignenvectors are $\begin{pmatrix} 1 \\ 0\\1 \end{pmatrix}$ , $\begin{pmatrix} 0 \\ 1 \\1\end{pmatrix}$, and $\begin{pmatrix} 1 \\ 1 \\1\end{pmatrix}$, respectively.

Answer: $\begin{pmatrix} 3&5&-5 \\ 5&3&-5 \\ 5&5&-7\end{pmatrix}$

I keep getting $\begin{pmatrix} 1&3&-3 \\ 5&3&-5 \\ 3&3&-5\end{pmatrix}$, so I am only getting the second row correct. I know that you're supposed to use the formula pA = PD$P^{-1}$. I had $\begin{pmatrix} 1&0&1 \\ 0&1&1 \\ 1&1&1\end{pmatrix}$ as $P$, $\begin{pmatrix} -2&0&0 \\ 0&-2&0 \\ 0&0&3\end{pmatrix}$ as $D$, and $\begin{pmatrix} 1&0&0 \\ -1&0&1 \\ 1&1&-1\end{pmatrix}$ as $P^{-1}$, and found the product to be $\begin{pmatrix} 1&3&-3 \\ 5&3&-5 \\ 3&3&-5\end{pmatrix}$ which is not right. Am I doing something wrong?

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    Your inverse is incorrect...you can check this yourself.2012-06-24
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    I checked it more than three times, and the elementary operations for both the inverse and the original matrix appears to be right for me...2012-06-24
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    But clearly those two matrices do NOT multiply to give the identity matrix. For a start the top left entry would be $2$.2012-06-24
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    Yeah, that's the problem. I don't get how the inverse is wrong though. I thought I derived it right2012-06-24
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    Oh, I think I got it. Thanks fretty!2012-06-24

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