Question:
Construct a $2 \times 2$ matrix $A=\left[a_{i j}\right]$ whose elements $a_{i j}$ are given by $a_{i j}= \begin{cases}\frac{|-3 i+j|}{2} & , \text { if } i \neq j \\ (i+j)^{2} & , \text { if } i=j\end{cases}$
Solution:
Given: $a_{i j}= \begin{cases}\frac{|-3 i+j|}{2}, & \text { if } i \neq j \\ (i+j)^{2}, & \text { if } i=j\end{cases}$
$a_{11}=(1+1)^{2}=4$
$a_{12}=\frac{|-3 \times 1+2|}{2}=\frac{1}{2}$
$a_{21}=\frac{|-3 \times 2+1|}{2}=\frac{5}{2}$
$a_{22}=(2+2)^{2}=16$
Hence, the matrix $A=\left[\begin{array}{cc}4 & \frac{1}{2} \\ \frac{5}{2} & 16\end{array}\right]$.