Constructing a Matrix using aij = 2i
Question:
Construct a \( 3 \times 4 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = 2i \).
Step 1: Matrix Order
- Rows → \( i = 1, 2, 3 \)
- Columns → \( j = 1, 2, 3, 4 \)
Step 2: Compute Elements
Since \( a_{ij} = 2i \), the value depends only on \( i \), not on \( j \).
For \( i = 1 \):
\[ a_{1j} = 2(1) = 2 \Rightarrow \text{Row 1: } 2,\;2,\;2,\;2 \]
For \( i = 2 \):
\[ a_{2j} = 2(2) = 4 \Rightarrow \text{Row 2: } 4,\;4,\;4,\;4 \]
For \( i = 3 \):
\[ a_{3j} = 2(3) = 6 \Rightarrow \text{Row 3: } 6,\;6,\;6,\;6 \]
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 \\ 6 & 6 & 6 & 6 \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 \\ 6 & 6 & 6 & 6 \end{bmatrix} \]