Question

diagonalize A= {{2,-1,-1},{1,4,1},{-1,-1,2}}
and compute A^2345

Answer 1

$A=\begin{pmatrix} 2 & -1 & -1 \\ 1 & 4 & 1 \\ -1 & -1 & 2\end{pmatrix} = \begin{pmatrix} 0 & -1 & -1 \\ 6 & 4 & 1 \\ 0 & -1 & 2\end{pmatrix} = \begin{pmatrix} 0 & -1 & -1 \\ 6 & 4 & 1 \\ 0 & 0 & 3\end{pmatrix}$ =

$\begin{pmatrix} 0 & -1 & -1 \\ 6 & 4 & 1 \\ 0 & 0 & 3\end{pmatrix} = \begin{pmatrix} 0 & -1 & -1 \\ 6 & 3 & 0 \\ 0 & 0 & 3\end{pmatrix} = \begin{pmatrix} 6 & 2 & -1 \\ 6 & 3 & 0 \\ 0 & 0 & 3\end{pmatrix}$ =

$= \begin{pmatrix} 6 & 2 & 0 \\ 6 & 3 & 0 \\ 0 & 0 & 3\end{pmatrix} = \begin{pmatrix} 2 & 2 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{pmatrix} = \begin{pmatrix} 2 & 0 & 0 \\ 0& 3 & 0 \\ 0 & 0 & 3\end{pmatrix}$

Since A is diagonal now, it is easy to find A to any power.

$A^{2345} =\begin{pmatrix} 2^{2345} & 0 & 0 \\ 0& 3^{2345} & 0 \\ 0 & 0 & 3^{2345}\end{pmatrix}$