Question

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

• A. $\left[\begin{array}{ccc}na& 0& 0\\ 0& nb& 0\\ 0& 0& nc\end{array}\right]$
• B. $\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$
• C. $\left[\begin{array}{ccc}{a}^n& 0& 0\\ 0& b^n& 0\\ 0& 0& c^n\end{array}\right]$
• D. $Noneofthese$

Answer 1

The correct option is C

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

Since $A^2=A.A=\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]=\left[\begin{array}{ccc}{a}^2& 0& 0\\ 0& b^2& 0\\ 0& 0& c^2\end{array}\right]$
And $A^3=\left[\begin{array}{ccc}{a}^3& 0& 0\\ 0& b^3& 0\\ 0& 0& c^3\end{array}\right],....\Rightarrow A^n=A^{n-1}.A=\left[\begin{array}{ccc}{a}^{n-1}& 0& 0\\ 0& b^{n-1}& 0\\ 0& 0& c^{n-1}\end{array}\right]\left[\begin{array}{ccc}a& 0& 0\\ 0& b& 0\\ 0& 0& c\end{array}\right]$
$=\left[\begin{array}{ccc}{a}^n& 0& 0\\ 0& b^n& 0\\ 0& 0& c^n\end{array}\right]$