📘 Question
If \(A = [a_{ij}]_{2 \times 2}\), where:
\[
a_{ij} =
\begin{cases}
1, & i \ne j \\
0, & i = j
\end{cases}
\]
Find \(A^2\).
✏️ Step-by-Step Solution
Step 1: Construct matrix
For \(2 \times 2\):
\[
A =
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\]
—
Step 2: Multiply \(A \cdot A\)
\[
A^2 =
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
\]
\[
=
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\]
—
✅ Final Answer
\[
\boxed{
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
}
\]
—
💡 Key Concept
This matrix swaps elements. Squaring it gives identity matrix:
\[
A^2 = I
\]