Question

Let A = $\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$, then Aⁿ is equal to

(a) $\left[\begin{array}{ccc}{a}^n& 0& 0\\ 0& a^n& 0\\ 0& 0& a\end{array}\right]$

(b) $\left[\begin{array}{ccc}{a}^n& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$

(c) $\left[\begin{array}{ccc}{a}^n& 0& 0\\ 0& a^n& 0\\ 0& 0& a^n\end{array}\right]$

(d) $\left[\begin{array}{ccc}na& 0& 0\\ 0& na& 0\\ 0& 0& na\end{array}\right]$

Answer 1

(c) $\left[\begin{array}{ccc}{a}^n& 0& 0\\ 0& a^n& 0\\ 0& 0& a^n\end{array}\right]$

$$\begin{gathered} \mathrm{Here}, \\ A=\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right] \\ \Rightarrow A^2=\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]=\left[\begin{array}{ccc}{a}^2& 0& 0\\ 0& a^2& 0\\ 0& 0& a^2\end{array}\right] \\ \Rightarrow A^3=\left[\begin{array}{ccc}{a}^2& 0& 0\\ 0& a^2& 0\\ 0& 0& a^2\end{array}\right]\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]=\left[\begin{array}{ccc}{a}^3& 0& 0\\ 0& a^3& 0\\ 0& 0& a^3\end{array}\right] \\ \mathrm{This}\mathrm{pattern}\mathrm{is}\mathrm{applicable}\mathrm{on}\mathrm{all}\mathrm{natural}\mathrm{numbers}. \\ \therefore A^n=\left[\begin{array}{ccc}{a}^n& 0& 0\\ 0& a^n& 0\\ 0& 0& a^n\end{array}\right] \end{gathered}$$