Question

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

• 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. $\left[\begin{array}{ccc}o& x^{100}& o\\ o& o& x^{100}\\ x^{100}& o& o\end{array}\right]$
• D. $\left[\begin{array}{ccc}o& x^{100}& o\\ x^{100}& o& o\\ o& o& x^{100}\end{array}\right]$

Answer 1

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

$A_n=\left\{\begin{array}{ccc}0& 0& x\\ 0& x& 0\\ x& 0& 0\end{array}\right\}{A}_{n+1}$
$A_{n+1}=\left\{\begin{array}{ccc}0& 0& x\\ 0& x& 0\\ x& 0& 0\end{array}\right\}$
$A=\left\{\begin{array}{ccc}0& 0& x\\ 0& x& 0\\ x& 0& 0\end{array}\right\}=-xI$
$A^{100}=x^{100}{I}^{100}$
$\therefore A^{100}=\left\{\begin{array}{ccc}{x}^{100}& 0& 0\\ 0& x^{100}& 0\\ 0& 0& x^{100}\end{array}\right\}$