Question #129572

diagonalize A= {{2,-1,-1},{1,4,1},{-1,-1,2}}

and compute A^2345

and compute A^2345

Expert's answer

"A=\\begin{pmatrix} 2 & -1 & -1 \\\\\n1 & 4 & 1 \\\\\n-1 & -1 & 2\\end{pmatrix} = \\begin{pmatrix} 0 & -1 & -1 \\\\\n6 & 4 & 1 \\\\\n0 & -1 & 2\\end{pmatrix} = \\begin{pmatrix} 0 & -1 & -1 \\\\\n6 & 4 & 1 \\\\\n0 & 0 & 3\\end{pmatrix}" =

"\\begin{pmatrix} 0 & -1 & -1 \\\\\n6 & 4 & 1 \\\\\n0 & 0 & 3\\end{pmatrix} = \\begin{pmatrix} 0 & -1 & -1 \\\\\n6 & 3 & 0 \\\\\n0 & 0 & 3\\end{pmatrix} = \\begin{pmatrix} 6 & 2 & -1 \\\\\n6 & 3 & 0 \\\\\n0 & 0 & 3\\end{pmatrix}" =

"= \\begin{pmatrix} 6 & 2 & 0 \\\\\n6 & 3 & 0 \\\\\n0 & 0 & 3\\end{pmatrix} = \\begin{pmatrix} 2 & 2 & 0 \\\\\n0 & 3 & 0 \\\\\n0 & 0 & 3\\end{pmatrix} = \\begin{pmatrix} 2 & 0 & 0 \\\\\n0& 3 & 0 \\\\\n0 & 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 \\\\\n0& 3^{2345} & 0 \\\\\n0 & 0 & 3^{2345}\\end{pmatrix}"

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