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