Constructing a Matrix using aij = 2i − j
Question:
Construct a \( 2 \times 3 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = 2i – j \).
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 \( a_{ij} = 2i – j \):
For \( i = 1 \):
\[ a_{11} = 2(1) – 1 = 1,\quad a_{12} = 2(1) – 2 = 0,\quad a_{13} = 2(1) – 3 = -1 \]
For \( i = 2 \):
\[ a_{21} = 2(2) – 1 = 3,\quad a_{22} = 2(2) – 2 = 2,\quad a_{23} = 2(2) – 3 = 1 \]
(These values are obtained by substituting row index \(i\) and column index \(j\) into the formula.)
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 1 & 0 & -1 \\ 3 & 2 & 1 \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 1 & 0 & -1 \\ 3 & 2 & 1 \end{bmatrix} \]