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} 

• \[\begin{bmatrix}a^n & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\]

• \[\begin{bmatrix}a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a^n\end{bmatrix}\]

•  \[\begin{bmatrix}na & 0 & 0 \\ 0 & na & 0 \\ 0 & 0 & na\end{bmatrix}\]

Answer 1

 \[\begin{bmatrix}a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a^n\end{bmatrix}\]

\[Here, \]

\[A = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\]

\[ \Rightarrow A^2 = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix} = \begin{bmatrix}a^2 & 0 & 0 \\ 0 & a^2 & 0 \\ 0 & 0 & a^2\end{bmatrix}\]

\[ \Rightarrow A^3 = \begin{bmatrix}a^2 & 0 & 0 \\ 0 & a^2 & 0 \\ 0 & 0 & a^2\end{bmatrix}\begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix} = \begin{bmatrix}a^3 & 0 & 0 \\ 0 & a^3 & 0 \\ 0 & 0 & a^3\end{bmatrix}\]

\[\]

This pattern is applicable on all natural numbers .

\[ \therefore A^n = \begin{bmatrix}a^n & 0 & 0 \\ 0 & a^n & 0 \\ 0 & 0 & a^n\end{bmatrix}\]