Constructing a Matrix using aij = ½|−3i + j|
Question:
Construct a \( 3 \times 4 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = \frac{1}{2}\left| -3i + j \right| \).
Step 1: Matrix Order
- Rows → \( i = 1, 2, 3 \)
- Columns → \( j = 1, 2, 3, 4 \)
Step 2: Compute Elements
For \( i = 1 \):
\[ a_{11} = \frac{1}{2}|-3 + 1| = 1,\quad a_{12} = \frac{1}{2}|-3 + 2| = \frac{1}{2},\quad a_{13} = \frac{1}{2}|-3 + 3| = 0,\quad a_{14} = \frac{1}{2}|-3 + 4| = \frac{1}{2} \]
For \( i = 2 \):
\[ a_{21} = \frac{1}{2}|-6 + 1| = \frac{5}{2},\quad a_{22} = \frac{1}{2}|-6 + 2| = 2,\quad a_{23} = \frac{1}{2}|-6 + 3| = \frac{3}{2},\quad a_{24} = \frac{1}{2}|-6 + 4| = 1 \]
For \( i = 3 \):
\[ a_{31} = \frac{1}{2}|-9 + 1| = 4,\quad a_{32} = \frac{1}{2}|-9 + 2| = \frac{7}{2},\quad a_{33} = \frac{1}{2}|-9 + 3| = 3,\quad a_{34} = \frac{1}{2}|-9 + 4| = \frac{5}{2} \]
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 1 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{5}{2} & 2 & \frac{3}{2} & 1 \\ 4 & \frac{7}{2} & 3 & \frac{5}{2} \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 1 & \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{5}{2} & 2 & \frac{3}{2} & 1 \\ 4 & \frac{7}{2} & 3 & \frac{5}{2} \end{bmatrix} \]