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# Construct a $2 \times 2$ matrix, $A=[a_{ij}],$whose elements are given by: $a_{ij}=\frac{(i+j)^2}{2}\qquad$

$\begin{array}{1 1} \begin{bmatrix}1 & \frac{9}{2}\\\frac{9}{2} & 8\end{bmatrix} \\\begin{bmatrix}2 & \frac{9}{2}\\\frac{9}{2} & 8\end{bmatrix} \\ \begin{bmatrix}2 & \frac{9}{3}\\\frac{9}{2} & 8\end{bmatrix} \\ \begin{bmatrix}2 & \frac{9}{2}\\\frac{9}{2} & -8\end{bmatrix} \end{array}$

Toolbox:
• In general $a_{2\times 2}$ matrix is given by$\begin{bmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{bmatrix}$
• Elements are given by $a_{ij}=\frac{(i+j)^2}{2}$, where (i, j) can be either (1,1), (1,2), (2,2) or (2,1)
Given, $a_{ij}=\frac{(i+j)^2}{2}, \Rightarrow$
$a_{11}=\frac{(1+1)^2}{2}=\frac{2^2}{2}=\frac{4}{2}=2.$
$a_{12}=\frac{(1+2)^2}{2}=\frac{3^2}{2}=\frac{9}{2}.$
$a_{21}=\frac{(2+1)^2}{2}=\frac{3^2}{2}=\frac{9}{2}.$
$a_{12}=\frac{(2+2)^2}{2}=\frac{4^2}{2}=\frac{16}{2}=8.$
Hence the required matrix is given by $A=\begin{bmatrix}2 & \frac{9}{2}\\\frac{9}{2} & 8\end{bmatrix}$