If A= 1&2 4&1 , then find A2 + 2A + 7I. - Vidyadip

Question

If $\text{A}=\begin{bmatrix}1&2\\4&1\end{bmatrix},$ then find A² + 2A + 7I.

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Answer

We have, $\text{A}=\begin{bmatrix}1&2\\4&1\end{bmatrix},$
$\Rightarrow\ \text{A}^2=\begin{bmatrix}1&2\\4&1\end{bmatrix}\begin{bmatrix}1&2\\4&1\end{bmatrix}$
$\Rightarrow\ \text{A}^2=\begin{bmatrix}1+8&2+2\\4+4&8+1\end{bmatrix}=\begin{bmatrix}9&4\\8&9\end{bmatrix}$
$\therefore\ \text{A}^2+2\text{A}+7$
$=\begin{bmatrix}9&4\\8&9\end{bmatrix}+\begin{bmatrix}2&4\\8&2\end{bmatrix}+\begin{bmatrix}7&0\\0&7\end{bmatrix}=\begin{bmatrix}18&8\\16&18\end{bmatrix}$

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