Constructing a Matrix using aij = (2i + j)2 / 2
Question:
Construct a \( 2 \times 2 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = \frac{(2i + j)^2}{2} \).
Step 1: Matrix Order
- Rows → \( i = 1, 2 \)
- Columns → \( j = 1, 2 \)
Step 2: Compute Elements
For \( i = 1 \):
\[ a_{11} = \frac{(2\cdot1 + 1)^2}{2} = \frac{3^2}{2} = \frac{9}{2},\quad a_{12} = \frac{(2\cdot1 + 2)^2}{2} = \frac{4^2}{2} = 8 \]
For \( i = 2 \):
\[ a_{21} = \frac{(2\cdot2 + 1)^2}{2} = \frac{5^2}{2} = \frac{25}{2},\quad a_{22} = \frac{(2\cdot2 + 2)^2}{2} = \frac{6^2}{2} = 18 \]
Step 3: Form the Matrix
\[ A = \begin{bmatrix} \frac{9}{2} & 8 \\ \frac{25}{2} & 18 \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} \frac{9}{2} & 8 \\ \frac{25}{2} & 18 \end{bmatrix} \]