<s>[INST] Computation of 10th power of a matrix I have been given the following matrix $A$. I am required to find 10th power of $A$.
$$
A = \begin{pmatrix}
 3 & 1 & 1 \\
 2 & 4 & 2 \\
 -1 & -1 & 1 \\
\end{pmatrix}
$$
I tried using Cayley Hamilton Theorem and found the characteristic polynomial as:
$t^3-8t^2+20t-16=0$.From here I find myself unable to compute the said power. Long division is daunting to me in this case. Please suggest how to go about it. The eigenvalues come out to be $2$( double) and $4$.
 [/INST] The eigenvectors for this matrix come out to be
$$\left( \begin{array}{c} -1\\1\\0\end{array} \right),\left( \begin{array}{c} -1\\0\\1\end{array} \right), \left( \begin{array}{c} -1\\-2\\1\end{array} \right)$$
To diagonalize the matrix, if $A = SDS^{-1}$, then
$$S = \left( \begin{array}{c c c} -1&-1&-1\\1&0&-2\\0&1&1\end{array} \right), A = \left( \begin{array}{c} 2&0&0\\0&2&0\\0&0&4\end{array} \right)$$
Then $A^{10} = SD^{10}S^{-1}$, and finding 10th power of a diagonal matrix is trivial.
Can you take it from here?
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