Question

If the matrix $A=\begin{pmatrix}2&0&0\\\ 0&2&0\\\ 2&0&2\end{pmatrix},$ then $A^{n}=\begin{pmatrix}a&0&0\\\ 0&a;&0\\\ b&0&a;\end{pmatrix}, n\,\in\,N$ where

• A. $a = 2n, b = 2^n$
• B. $a = 2^n, b = 2n$
• C. $a = 2^n, b = n^{2n-1}$
• D. $a = 2^n, b = n2^n$

Answer 1

Answer: (D)

$a = 2^n, b = n2^n$

Answer 2

$A=2\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 1&0&1\end{pmatrix} \Rightarrow A^{n}=2^{n}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ n&0&1\end{pmatrix}$