Constructing a Matrix using aij = i + j
Question:
Construct a \( 2 \times 3 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = i + j \).
Step 1: Matrix Order
A \( 2 \times 3 \) matrix has:
- 2 rows → \( i = 1, 2 \)
- 3 columns → \( j = 1, 2, 3 \)
Step 2: Compute Elements
Using the formula \( a_{ij} = i + j \):
For \( i = 1 \):
\[ a_{11} = 1 + 1 = 2,\quad a_{12} = 1 + 2 = 3,\quad a_{13} = 1 + 3 = 4 \]
For \( i = 2 \):
\[ a_{21} = 2 + 1 = 3,\quad a_{22} = 2 + 2 = 4,\quad a_{23} = 2 + 3 = 5 \]
(Each element is obtained by adding its row index \(i\) and column index \(j\).)
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix} \]