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Question
Let A = \[\begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\], then An is equal to
Options
\begin{bmatrix}a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a\end{bmatrix}
\[\begin{bmatrix}a^n & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\]
\[\begin{bmatrix}a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a^n\end{bmatrix}\]
\[\begin{bmatrix}na & 0 & 0 \\ 0 & na & 0 \\ 0 & 0 & na\end{bmatrix}\]
Solution
\[\begin{bmatrix}a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a^n\end{bmatrix}\]
\[Here, \]
\[A = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix} = \begin{bmatrix}a^2 & 0 & 0 \\ 0 & a^2 & 0 \\ 0 & 0 & a^2\end{bmatrix}\]
\[ \Rightarrow A^3 = \begin{bmatrix}a^2 & 0 & 0 \\ 0 & a^2 & 0 \\ 0 & 0 & a^2\end{bmatrix}\begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix} = \begin{bmatrix}a^3 & 0 & 0 \\ 0 & a^3 & 0 \\ 0 & 0 & a^3\end{bmatrix}\]
\[\]
This pattern is applicable on all natural numbers .
\[ \therefore A^n = \begin{bmatrix}a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a^n\end{bmatrix}\]
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