Question

If $A=\left[\begin{array}{ccc}0& 0& x\\ 0& x& 0\\ x& 0& 0\end{array}\right]$, then $A^{100}=$

• A. $$\left[\begin{array}{ccc}0& 0& x^{100}\\ 0& x^{100}& 0\\ x^{100}& 0& 0\end{array}\right]$$
• B. $$\left[\begin{array}{ccc}{x}^{100}& 0& 0\\ 0& x^{100}& 0\\ 0& 0& x^{100}\end{array}\right]$$
• C. $$\left[\begin{array}{ccc}0& x^{100}& 0\\ 0& 0& x^{100}\\ x^{100}& 0& 0\end{array}\right]$$
• D. $$\left[\begin{array}{ccc}0& x^{100}& 0\\ x^{100}& 0& 0\\ 0& 0& x^{100}\end{array}\right]$$

Answer 1

The correct option is B $$\left[\begin{array}{ccc}{x}^{100}& 0& 0\\ 0& x^{100}& 0\\ 0& 0& x^{100}\end{array}\right]$$

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

$A^3=A^2\cdot A=\left[\begin{array}{ccc}{x}^2& 0& 0\\ 0& x^2& 0\\ 0& 0& x^2\end{array}\right]\cdot \left[\begin{array}{ccc}0& 0& x\\ 0& x& 0\\ x& 0& 0\end{array}\right]=\left[\begin{array}{ccc}0& 0& x^3\\ 0& x^3& 0\\ x^3& 0& 0\end{array}\right]$
Similarly $A^4=\left[\begin{array}{ccc}{x}^4& 0& 0\\ 0& x^4& 0\\ 0& 0& x^4\end{array}\right]$
and $A^5=\left[\begin{array}{ccc}0& 0& x^5\\ 0& x^5& 0\\ x^5& 0& 0\end{array}\right]$ so on.

$\Rightarrow A^{100}=\left[\begin{array}{ccc}{x}^{100}& 0& 0\\ 0& x^{100}& 0\\ 0& 0& x^{100}\end{array}\right]$