📘 Question
If
\[
A =
\begin{bmatrix}
i & 0 \\
0 & i
\end{bmatrix}
= iI
\]
Find \(A^{4n}\), where \(n \in \mathbb{N}\).
✏️ Step-by-Step Solution
Step 1: Express matrix
\[
A = iI
\]
—
Step 2: Use power rule
\[
A^{4n} = (iI)^{4n} = i^{4n} \cdot I^{4n}
\]
—
Step 3: Simplify
\[ I^{4n} = I \]
\[ i^4 = 1 \Rightarrow i^{4n} = 1 \]
—Step 4: Final result
\[
A^{4n} = 1 \cdot I = I
\]
—
✅ Final Answer
\[
\boxed{I}
\]
—
💡 Key Concept
- \(i^4 = 1\)
- \(I^n = I\)
- \((kI)^n = k^n I\)