Constructing a Matrix using aij = i × j
Question:
Construct a \( 2 \times 3 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = i \times j \).
Step 1: Matrix Order
A \( 2 \times 3 \) matrix has:
- 2 rows → \( i = 1, 2 \)
- 3 columns → \( j = 1, 2, 3 \)
So we compute each element using the rule \( a_{ij} = i \times j \). :contentReference[oaicite:0]{index=0}
Step 2: Compute Elements
For first row \( (i = 1) \):
\[ a_{11} = 1 \times 1 = 1,\quad a_{12} = 1 \times 2 = 2,\quad a_{13} = 1 \times 3 = 3 \]
For second row \( (i = 2) \):
\[ a_{21} = 2 \times 1 = 2,\quad a_{22} = 2 \times 2 = 4,\quad a_{23} = 2 \times 3 = 6 \]
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{bmatrix} \]