Question

Let $A=\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]$, then $A^n$ 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

Answer: (C)

$A^2=\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]=\left[\begin{array}{ccc}{a}^2& 0& 0\\ 0& a^2& 0\\ 0& 0& a^2\end{array}\right]$
$A^3=\left[\begin{array}{ccc}{a}^2& 0& 0\\ 0& a^2& 0\\ 0& 0& a^2\end{array}\right]\left[\begin{array}{ccc}a& 0& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right]=\left[\begin{array}{ccc}{a}^3& 0& 0\\ 0& a^3& 0\\ 0& 0& a^3\end{array}\right]$