Constructing a Matrix using aij = j
Question:
Construct a \( 3 \times 4 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = j \).
Step 1: Matrix Order
- Rows → \( i = 1, 2, 3 \)
- Columns → \( j = 1, 2, 3, 4 \)
Step 2: Compute Elements
Since \( a_{ij} = j \), the value depends only on the column index \( j \), not on \( i \).
For each row:
\[ a_{i1} = 1,\quad a_{i2} = 2,\quad a_{i3} = 3,\quad a_{i4} = 4 \]
So every row will be the same.
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{bmatrix} \]