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

(A) Since it is a $2 \times 2$ matrix,it has $2$ rows and $2$ columns.
Let the matrix be $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$.
Given $a_{ij} = \frac{(i+j)^2}{2}$,we calculate each element:

$a_{11} = \frac{(1+1)^2}{2} = \frac{4}{2} = 2$	$a_{12} = \frac{(1+2)^2}{2} = \frac{9}{2}$
$a_{21} = \frac{(2+1)^2}{2} = \frac{9}{2}$	$a_{22} = \frac{(2+2)^2}{2} = \frac{16}{2} = 8$

Thus,the required matrix $A$ is $A = \begin{bmatrix} 2 & \frac{9}{2} \\ \frac{9}{2} & 8 \end{bmatrix}$.