Constructing a Matrix using aij = (i + j)2 / 2
Question:
Construct a \( 2 \times 3 \) matrix \( A = [a_{ij}] \) whose elements are given by \( a_{ij} = \frac{(i + j)^2}{2} \).
Concept
In matrix notation, each element is determined using its row index \(i\) and column index \(j\).
Step 1: Matrix Order
- Rows → \( i = 1, 2 \)
- Columns → \( j = 1, 2, 3 \)
Step 2: Compute Elements
For \( i = 1 \):
\[ a_{11} = \frac{(1+1)^2}{2} = \frac{4}{2} = 2,\quad a_{12} = \frac{(1+2)^2}{2} = \frac{9}{2},\quad a_{13} = \frac{(1+3)^2}{2} = \frac{16}{2} = 8 \]
For \( i = 2 \):
\[ a_{21} = \frac{(2+1)^2}{2} = \frac{9}{2},\quad a_{22} = \frac{(2+2)^2}{2} = \frac{16}{2} = 8,\quad a_{23} = \frac{(2+3)^2}{2} = \frac{25}{2} \]
(Each entry is calculated by substituting values of \(i\) and \(j\) into the formula.)
Step 3: Form the Matrix
\[ A = \begin{bmatrix} 2 & \frac{9}{2} & 8 \\ \frac{9}{2} & 8 & \frac{25}{2} \end{bmatrix} \]
Final Answer
\[ A = \begin{bmatrix} 2 & \frac{9}{2} & 8 \\ \frac{9}{2} & 8 & \frac{25}{2} \end{bmatrix} \]