Question:
Construct a $4 \times 3$ matrix whose elements are given by $a_{i j}=\frac{i}{j}$.
Solution:
It is (4 x 3) matrix. So it has 4 rows and 3 columns
Given $a_{i j}=\frac{i}{j}$
So, $a_{11}=1, a_{12}=\frac{1}{2}, a_{13}=\frac{1}{3}$
$a_{21}=2, a_{22}=1, a_{23}=\frac{2}{3}$
$a_{31}=3 \cdot a_{32}=\frac{3}{2}, a_{33}=1$
$a_{41}=4 \cdot a_{42}=2 \cdot a_{43}=\frac{4}{3}$
So, the matrix $=\left[\begin{array}{ccc}1 & \frac{1}{2} & \frac{1}{3} \\ 2 & 1 & \frac{2}{3} \\ 3 & \frac{3}{2} & 1 \\ 4 & 2 & \frac{4}{3}\end{array}\right]$
Conclusion: Therefore, Matrix is $\left[\begin{array}{ccc}1 & \frac{1}{2} & \frac{1}{3} \\ 2 & 1 & \frac{2}{3} \\ 3 & \frac{3}{2} & 1 \\ 4 & 2 & \frac{4}{3}\end{array}\right]$