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