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 \( i = 1 \):
\[ a_{11}=2,\; a_{12}=3,\; a_{13}=4,\; a_{14}=5 \]
For \( i = 2 \):
\[ a_{21}=3,\; a_{22}=4,\; a_{23}=5,\; a_{24}=6 \]
For \( i = 3 \):
\[ a_{31}=4,\; a_{32}=5,\; a_{33}=6,\; a_{34}=7 \]
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \end{bmatrix} \]