Question

If $P=\left(\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right)$ then $P^5$ equals

• A. $P$
• B. $2P$
• C. $-P$
• D. $-2P$

Answer 1

The correct option is A $P$

$P=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$

$P^2=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$

$=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]=P$

Now,

$P^2=P{P}^5=P^2\times P^3\Rightarrow P^5=P^4$

$(\because P^2=P)\Rightarrow P^5=P^2\times P^2$

$P^2=P\times P$

$\Rightarrow P^5=P^2=P$

$\therefore P^5=P$

$\therefore$Option A is correct