Question:
Construct a $3 \times 4$ matrix whose elements are given by $a_{i j}=\frac{1}{2}|-3 i+j|$.
Solution:
It is a (3 x 4) matrix. So, it has 3 rows and 4 columns.
Given $a_{i j}=\frac{|-3 i+j|}{2}$
So, $a_{11}=1, a_{12}=\frac{1}{2}, a_{13}=0, a_{13}=\frac{1}{2}$,
$a_{21}=\frac{5}{2}, a_{22}=2, a_{23}=\frac{3}{2}, a_{13}=1$
$a_{31}=4, a_{32}=\frac{7}{2}, a_{33}=3, a_{13}=\frac{5}{2}$
So, the matrix $=\left[\begin{array}{cccc}1 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{5}{2} & 2 & \frac{3}{2} & 1 \\ 4 & \frac{7}{2} & 3 & \frac{5}{2}\end{array}\right]$
Conclusion: Therefore, Matrix is $\left[\begin{array}{cccc}1 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{5}{2} & 2 & \frac{3}{2} & 1 \\ 4 & \frac{7}{2} & 3 & \frac{5}{2}\end{array}\right]$