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

Toolbox:
• If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: $\begin{bmatrix}AB\end{bmatrix}_{i,j} = A_{i,1}B_{1,j} + A_{i,2}B_{2,j} + A_{i,3}B_{3,j} ... A_{i,n}B_{n,j}$
• The scalar multiplication $cA$ of a matrix $A$ and a number $c$ (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of $A$ by $c$.
• The sum / difference $A(+/-)B$ of two $m$-by-$n$ matrices $A$ and $B$ is calculated entrywise: $(A (+/-) B)_{i,j} = A_{i,j} +/- B_{i,j}$ where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
Step1:
Given
$A=\begin{bmatrix}1 & 2\\4 & 1\end{bmatrix}$
$A^2=A.A\Rightarrow \begin{bmatrix}1 & 2\\4 & 1\end{bmatrix}\begin{bmatrix}1 & 2\\4 & 1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}1(1)+2(4) & 1(2)+2(1)\\4(1)+1(4) & 4(2)+1(1)\end{bmatrix}$
$\Rightarrow \begin{bmatrix}1+8& 2+2\\4+4 & 8+1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}9& 4\\8 & 9\end{bmatrix}$
Step2:
$A^2+2A+7I=\begin{bmatrix}9 & 4\\8 & 9\end{bmatrix}+2\begin{bmatrix}1 & 2\\4 & 1\end{bmatrix}+7\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}9 & 4\\8 & 9\end{bmatrix}+\begin{bmatrix}2 & 4\\8 & 2\end{bmatrix}+\begin{bmatrix}7 & 0\\0 & 7\end{bmatrix}$
$\Rightarrow \begin{bmatrix}9+2+7 & 4+4+0\\8+8+0 & 9+2+7\end{bmatrix}$
$\Rightarrow \begin{bmatrix}18 & 8\\16 & 18\end{bmatrix}$