Answer to Question #129572 in Electrical Engineering for Hash

Question #129572
diagonalize A= {{2,-1,-1},{1,4,1},{-1,-1,2}}
and compute A^2345
1
Expert's answer
2020-09-03T08:27:12-0400

A=(211141112)=(011641012)=(011641003)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} =

(011641003)=(011630003)=(621630003)\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} =

=(620630003)=(220030003)=(200030003)= \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.

A2345=(223450003234500032345)A^{2345} =\begin{pmatrix} 2^{2345} & 0 & 0 \\ 0& 3^{2345} & 0 \\ 0 & 0 & 3^{2345}\end{pmatrix}


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