LA2.1.3

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If A={\begin{bmatrix}2&3\\-1&2\end{bmatrix}}\, Show that A^{2}-4A+7I=O\,

A^{2}={\begin{bmatrix}2&3\\-1&2\end{bmatrix}}{\begin{bmatrix}2&3\\-1&2\end{bmatrix}}\,

{\begin{bmatrix}4-3&6+6\\-2-2&-3+4\end{bmatrix}}={\begin{bmatrix}1&12\\-4&1\end{bmatrix}}\,

-4A=-4{\begin{bmatrix}2&3\\-1&2\end{bmatrix}}={\begin{bmatrix}-8&-12\\4&-8\end{bmatrix}}\,

Therefore A^{2}-4A+7I={\begin{bmatrix}1&12\\-4&1\end{bmatrix}}+{\begin{bmatrix}-8&-12\\4&-8\end{bmatrix}}+{\begin{bmatrix}7&0\\0&7\end{bmatrix}}\,

{\begin{bmatrix}1-8+7&12-12+0\\-4+4+0&1-8+7\end{bmatrix}}={\begin{bmatrix}0&0\\0&0\end{bmatrix}}=O\,

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