📘 Question
Construct a \(2 \times 2\) matrix \(A = [a_{ij}]\), where:
\[
a_{ij} =
\begin{cases}
\frac{|-3i + j|}{2}, & i \ne j \\
(i + j)^2, & i = j
\end{cases}
\]
✏️ Step-by-Step Solution
For a \(2 \times 2\) matrix, \(i, j = 1, 2\)
Step 1: Find each element
- \(a_{11}\) (i = j): \[ (1 + 1)^2 = 4 \]
- \(a_{12}\) (i ≠ j): \[ \frac{| -3(1) + 2 |}{2} = \frac{|-3 + 2|}{2} = \frac{1}{2} \]
- \(a_{21}\) (i ≠ j): \[ \frac{| -3(2) + 1 |}{2} = \frac{|-6 + 1|}{2} = \frac{5}{2} \]
- \(a_{22}\) (i = j): \[ (2 + 2)^2 = 16 \]
Step 2: Form the matrix
\[
A =
\begin{bmatrix}
4 & \frac{1}{2} \\
\frac{5}{2} & 16
\end{bmatrix}
\]
✅ Final Answer
\[
\boxed{
\begin{bmatrix}
4 & \frac{1}{2} \\
\frac{5}{2} & 16
\end{bmatrix}
}
\]
💡 Key Concept
To construct a matrix from \(a_{ij}\), substitute values of \(i\) and \(j\) one by one. Use different formulas depending on whether \(i = j\) or \(i \ne j\).