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