Find A2 - 5A + 6I if A = 2&0&1 2&1&3 1&-1&0 . - Vidyadip

Question

Find A² - 5A + 6I if A = $\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}.$

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Answer

$\text{A}^2-5\text{A}+6\text{I}=\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}-5\begin{bmatrix}2&0&1\\2&1&3\\1&-1&0\end{bmatrix}+6\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$
$=\begin{bmatrix}4 + 0 + 1&0 + 0 - 1&2 + 0 + 0\\4 + 2 + 3&0 + 1 - 3&2 + 3 + 0\\2 - 2 + 0&0 - 1 - 0&1 - 3 + 0\end{bmatrix} - \begin{bmatrix}10&0&5\\10&5&15\\5&-5&0\end{bmatrix} + \begin{bmatrix}6&0&0\\0&6&0\\0&0&6\end{bmatrix}$
$=\begin{bmatrix}5&-1&2\\9&-2&5\\0&-1&-2\end{bmatrix}-\begin{bmatrix}10&0&5\\10&5&15\\5&-5&0\end{bmatrix} + \begin{bmatrix}6&0&0\\0&6&0\\0&0&6\end{bmatrix} $
$= \begin{bmatrix}5 - 10 + 6& -1-0 + 0&2 - 5 + 0\\9 - 10 + 0&-2 - 5 + 6&5-15 + 0\\0-5 + 0&-1 + 5+0&-2-0 + 6\end{bmatrix}$
$=\begin{bmatrix}1&-1&-3\\-1&-1&-10\\-5&4&4\end{bmatrix}$

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