Constructing a Matrix using aij = 2i + i/j
Question:
Construct a \( 4 \times 3 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = 2i + \frac{i}{j} \).
Step 1: Matrix Order
- Rows → \( i = 1, 2, 3, 4 \)
- Columns → \( j = 1, 2, 3 \)
Step 2: Compute Elements
For \( i = 1 \):
\[ a_{11} = 2(1) + \frac{1}{1} = 3,\quad a_{12} = 2(1) + \frac{1}{2} = \frac{5}{2},\quad a_{13} = 2(1) + \frac{1}{3} = \frac{7}{3} \]
For \( i = 2 \):
\[ a_{21} = 2(2) + \frac{2}{1} = 6,\quad a_{22} = 2(2) + \frac{2}{2} = 5,\quad a_{23} = 2(2) + \frac{2}{3} = \frac{14}{3} \]
For \( i = 3 \):
\[ a_{31} = 2(3) + \frac{3}{1} = 9,\quad a_{32} = 2(3) + \frac{3}{2} = \frac{15}{2},\quad a_{33} = 2(3) + \frac{3}{3} = 7 \]
For \( i = 4 \):
\[ a_{41} = 2(4) + \frac{4}{1} = 12,\quad a_{42} = 2(4) + \frac{4}{2} = 10,\quad a_{43} = 2(4) + \frac{4}{3} = \frac{28}{3} \]
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 3 & \frac{5}{2} & \frac{7}{3} \\ 6 & 5 & \frac{14}{3} \\ 9 & \frac{15}{2} & 7 \\ 12 & 10 & \frac{28}{3} \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 3 & \frac{5}{2} & \frac{7}{3} \\ 6 & 5 & \frac{14}{3} \\ 9 & \frac{15}{2} & 7 \\ 12 & 10 & \frac{28}{3} \end{bmatrix} \]