Question

If $A=\left[\begin{array}{ccc}0& 0& x\\ 0& x& 0\\ x& 0& 0\end{array}\right]$ then find $A^{2n+1}$.

• A. $\left[\begin{array}{ccc}{x}^{2n+1}& 0& 0\\ 0& x^{2n-1}& 0\\ 0& 0& x^{2n+1}\end{array}\right]$
• B. $\left[\begin{array}{ccc}0& 0& x^{2n+1}\\ 0& x^{2n+1}& \\ x^{2n+1}& 0& 0\end{array}\right]$
• C. $\left[\begin{array}{ccc}{x}^n& 0& 0\\ 0& x^n& 0\\ 0& 0& x^n\end{array}\right]$
• D. $\left[\begin{array}{ccc}0& 0& x^n\\ 0& x^n& 0\\ x^n& 0& 0\end{array}\right]$

Answer 1

Answer: (B)

$\left[\begin{array}{ccc}0& 0& x^{2n+1}\\ 0& x^{2n+1}& \\ x^{2n+1}& 0& 0\end{array}\right]$

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

$A^4=A^2{A}^2=\left[\begin{array}{ccc}{x}^2& 0& 0\\ 0& x^2& 0\\ 0& 0& x^2\end{array}\right]\left[\begin{array}{ccc}{x}^2& 0& 0\\ 0& x^2& 0\\ 0& 0& x^2\end{array}\right]=\left[\begin{array}{ccc}{x}^4& 0& 0\\ 0& x^4& 0\\ 0& 0& x^4\end{array}\right]$
So, in general we can say that

$A^{2n}=\left[\begin{array}{ccc}{x}^{2n}& 0& 0\\ 0& x^{2n}& 0\\ 0& 0& x^{2n}\end{array}\right]$
Now, $A^{2n+1}=A^{2n}A\left[\begin{array}{ccc}{x}^{2n}& 0& 0\\ 0& x^{2n}& 0\\ 0& 0& x^{2n}\end{array}\right]\left[\begin{array}{ccc}0& 0& x\\ 0& x& 0\\ x& 0& 0\end{array}\right]=\left[\begin{array}{ccc}0& 0& x^{2n+1}\\ 0& x^{2n+1}& 0\\ x^{2n+1}& 0& 0\end{array}\right]$