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QUESTION

Find the eigenvalues and eigenvectors of the following matrices.
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$\left[\begin{array}{cc}3&4\\-2&-3\end{array}\right] \hspace{1cm}
\left[\begin{array}{cc}3&5\\-5&-3\end{array}\right]$
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For each of these matrices $A$ write down where possible the
matrix $M$ such that $M^{-1}AM$ is diagonal and check that your
$M$ works.

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ANSWER



\begin{tabular}{lclc}
For the first matrix\\
Eigenvalue&1&Eigenvector&$\left[\begin{array}{c}2\\-1\end{array}\right]$\\
Eigenvalue&-1&Eigenvector&$\left[\begin{array}{c}1\\-1\end{array}\right]$\\
\\
For the second matrix\\

Eigenvalue&$\pm4i$&Eigenvector&$\left[\begin{array}{c}-5\\3\mp4i\end{array}\right]$\\

\end{tabular}

Where there are two independent eigenvectors the matrix $M$ which
has the eigenvectors as its columns will do.


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