Find ${A^2} - 5A + 6I,$If $A = \left[ {\begin{array}{cccccccccccccccccccc}2&0&1\\2&1&3\\1&{ - 1}&0\end{array}} \right]$

.:

We are given that $A = \left[ {\begin{array}{cccccccccccccccccccc}2&0&1\\2&1&3\\1&{ - 1}&0\end{array}} \right]$

$\Rightarrow$ ${A^2} = A \cdot A = \left[ {\begin{array}{cccccccccccccccccccc}2&0&1\\2&1&3\\1&{ - 1}&0\end{array}} \right]\left[ {\begin{array}{cccccccccccccccccccc}2&0&1\\2&1&3\\1&{ - 1}&0\end{array}} \right]$

$= \left[ {\begin{array}{cccccccccccccccccccc}{2 \times 2 + 0 \times 2 + 1 \times 1}&{2 \times 0 + 0 \times 1 + 1 \times ( - 1)}&{2 \times 1 + 0 \times 3 + 1 \times 0}\\{2 \times 2 + 1 \times 2 + 3 \times 1}&{2 \times 0 + 1 \times 1 + 3 \times ( - 1)}&{2 \times 1 + 1 \times 3 + 3 \times 0}\\{1 \times 2 + 2 \times ( - 1) + 0 \times 1}&{1 \times 0 + ( - 1) \times 1 + 0 \times ( - 1)}&{1 \times 1 + ( - 1) \times 3 + 0 \times 0}\end{array}} \right]$

= $\left[ {\begin{array}{cccccccccccccccccccc}{4 + 0 + 1}&{0 + 0 - 1}&{2 + 0 + 0}\\{4 + 2 + 3}&{0 + 1 - 3}&{2 + 3 + 0}\\{2 - 2 + 0}&{0 - 1 + 0}&{1 - 3 + 0}\end{array}} \right]$

$= \left[ {\begin{array}{cccccccccccccccccccc}5&{ - 1}&2\\9&{ - 2}&5\\0&{ - 1}&{ - 2}\end{array}} \right]$
$5A = 5\left[ {\begin{array}{cccccccccccccccccccc}2&0&1\\2&1&3\\1&{ - 1}&0\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{10}&0&5\\{10}&5&{15}\\5&{ - 5}&0\end{array}} \right]$

$6I = 6\left[ {\begin{array}{cccccccccccccccccccc}1&0&0\\0&1&0\\0&0&1\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}6&0&0\\0&6&0\\0&0&6\end{array}} \right]$

$\therefore$ ${A^2} - 5A + 6I = \left[ {\begin{array}{cccccccccccccccccccc}5&{ - 1}&2\\9&{ - 2}&5\\0&{ - 1}&{ - 2}\end{array}} \right] - \left[ {\begin{array}{cccccccccccccccccccc}{10}&0&5\\{10}&5&{15}\\5&{ - 5}&0\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}6&0&0\\0&6&0\\0&0&6\end{array}} \right]$

$= \left[ {\begin{array}{cccccccccccccccccccc}{ - 5}&{ - 1}&{ - 3}\\{ - 1}&{ - 7}&{ - 10}\\{ - 5}&4&{ - 2}\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}6&0&0\\0&6&0\\0&0&6\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}1&{ - 1}&{ - 3}\\{ - 1}&{ - 1}&{ - 10}\\{ - 5}&4&4\end{array}} \right]$