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