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QUESTION

Find the eigenvalues and eigenvectors of the following matrices.

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

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