📘 Question
Let
\[
A =
\begin{bmatrix}
a & 0 & 0 \\
0 & a & 0 \\
0 & 0 & a
\end{bmatrix}
\]
Find \(A^n\).
✏️ Step-by-Step Solution
Step 1: Recognize the matrix
This is a scalar matrix:
\[
A = aI
\]
Step 2: Use power property
\[ A^n = (aI)^n = a^n I^n \]
Since:
\[
I^n = I
\]
Step 3: Final result
\[
A^n = a^n I
\]
\[
=
\begin{bmatrix}
a^n & 0 & 0 \\
0 & a^n & 0 \\
0 & 0 & a^n
\end{bmatrix}
\]
✅ Final Answer
\[
\boxed{
\begin{bmatrix}
a^n & 0 & 0 \\
0 & a^n & 0 \\
0 & 0 & a^n
\end{bmatrix}
}
\]
💡 Key Concept
A scalar matrix behaves like a number multiplied by identity. So powers apply directly:
\[
(aI)^n = a^n I
\]