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QUESTION

Find the eigenvalues and eigenvectors of the following matrices.

$\left[\begin{array}{ccc}6&1&1\\1&6&1\\3&3&6\end{array}\right]
\hspace{1.5cm}
\left[\begin{array}{ccc}2&-1&-1\\0&3&2\\-1&1&2\end{array}\right]
\hspace{1.5cm}
\left[\begin{array}{ccc}-1&-1&0\\0&-1&-4\\1&0&-4\end{array}\right]$

For each matrix $A$ write down where possible a matrix $M$ such
that $M^{-1}AM$ is diagonal and check that $M$ works.

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ANSWER\\
\begin{tabular}{lcll}
First matrix\\
Eigenvalue&4&Eigenvector&$\left[\begin{array}{c}1\\1\\-3\end{array}\right]$\\
Eigenvalue&5&Eigenvector&$\left[\begin{array}{c}1\\-1\\0\end{array}\right]$\\
Eigenvalue&9&Eigenvector&$\left[\begin{array}{c}1\\1\\2\end{array}\right]$\\
\\
Second matrix\\
Eigenvalue&1&Eigenvector&$\left[\begin{array}{c}0\\1\\-1\end{array}\right]$\\
Eigenvalue&$3+\sqrt{2}$&Eigenvector&$\left[\begin{array}{c}-1\\\sqrt{2}\\1\end{array}\right]$\\
Eigenvalue&$3-\sqrt{2}$&Eigenvector&$\left[\begin{array}{c}-1\\-\sqrt{2}\\1\end{array}\right]$\\
\\
Third matrix\\
Eigenvalue&0&Eigenvector&$\left[\begin{array}{c}4\\-4\\1\end{array}\right]$\\
Eigenvalue&-3,-3&Eigenvector&$\left[\begin{array}{c}1\\2\\1\end{array}\right]$
only
\\
\end{tabular}

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



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