Question

If $P = \begin{pmatrix}2&-2&-4\\ -1&3&4\\ 1&-2&-3\end{pmatrix}$ then $P^{5}$ equals to

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

Answer 1

Answer: (A)

Given, $ P=\begin{pmatrix}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{pmatrix} $
$ P^{2} =P \cdot P=\begin{pmatrix}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{pmatrix} \begin{pmatrix}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{pmatrix} $
$ =\begin{pmatrix}4+2-4 & -4-6+8 & -8-8+12 \\ -2-3+4 & 2+9-8 & 4+12-12 \\ 2+2-3 & -2-6+6 & -4-8+9\end{pmatrix}$
$=\begin{pmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{pmatrix} =P $
$ \therefore P^{4} =P^{2}=P $
$ \Rightarrow P^{5} =P^{2}=P $