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