Constructing a Matrix using aij = (i − j)/(i + j)
Question:
Construct a \( 4 \times 3 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = \frac{i – j}{i + j} \).
Step 1: Matrix Order
- Rows → \( i = 1, 2, 3, 4 \)
- Columns → \( j = 1, 2, 3 \)
Step 2: Compute Elements
For \( i = 1 \):
\[ a_{11} = \frac{1-1}{1+1} = 0,\quad a_{12} = \frac{1-2}{1+2} = -\frac{1}{3},\quad a_{13} = \frac{1-3}{1+3} = -\frac{1}{2} \]
For \( i = 2 \):
\[ a_{21} = \frac{2-1}{2+1} = \frac{1}{3},\quad a_{22} = \frac{2-2}{2+2} = 0,\quad a_{23} = \frac{2-3}{2+3} = -\frac{1}{5} \]
For \( i = 3 \):
\[ a_{31} = \frac{3-1}{3+1} = \frac{1}{2},\quad a_{32} = \frac{3-2}{3+2} = \frac{1}{5},\quad a_{33} = \frac{3-3}{3+3} = 0 \]
For \( i = 4 \):
\[ a_{41} = \frac{4-1}{4+1} = \frac{3}{5},\quad a_{42} = \frac{4-2}{4+2} = \frac{1}{3},\quad a_{43} = \frac{4-3}{4+3} = \frac{1}{7} \]
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 0 & -\frac{1}{3} & -\frac{1}{2} \\ \frac{1}{3} & 0 & -\frac{1}{5} \\ \frac{1}{2} & \frac{1}{5} & 0 \\ \frac{3}{5} & \frac{1}{3} & \frac{1}{7} \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 0 & -\frac{1}{3} & -\frac{1}{2} \\ \frac{1}{3} & 0 & -\frac{1}{5} \\ \frac{1}{2} & \frac{1}{5} & 0 \\ \frac{3}{5} & \frac{1}{3} & \frac{1}{7} \end{bmatrix} \]