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**I-Think** The matrix $\displaystyle A=\left(\begin{array}{ccc}3&-1&0\\-4&-6&-6\\5&11&10\end{array}\right)$

has eigenvectors $\displaystyle \left(\begin{array}{c}1\\-1\\1\end{array}\right)$, $\displaystyle \left(\begin{array}{c}1\\2\\-3\end{array}\right)$ and $\displaystyle \left(\begin{array}{c}1\\1\\-2\end{array}\right)$ with corresponding eigenvalues 4, 1 and 2 respectively.

The matrix B has eigenvalues 2, 3, 1 with corresponding eigenvectors

$\displaystyle \left(\begin{array}{c}1\\-1\\1\end{array}\right)$, $\displaystyle \left(\begin{array}{c}1\\2\\-3\end{array}\right)$ and $\displaystyle \left(\begin{array}{c}1\\1\\-2\end{array}\right)$

Find a matrix **P **and a diagonal matrix **D** such that

$\displaystyle (A+B)^4=PDP^{-1}$

Not required to evaluate $\displaystyle P^{-1}$

A start and some hints would be very nice.