Find the matrix

Algebra Level 3

$P= \begin{bmatrix} \cos\cfrac { \pi }{ 4 } & -\sin\cfrac { \pi }{ 4 } \\ \sin\cfrac { \pi }{ 4 } & \cos\cfrac { \pi }{ 4 } \end{bmatrix} \qquad \qquad X = \left[ \begin{matrix} \cfrac { 1 }{ \sqrt { 2 } } \\ \cfrac { 1 }{ \sqrt { 2 } } \end{matrix} \right]$

For matrices $P$ and $X$ as defined above, find $P^3X$ .

\left[ \begin{matrix} 0 \\ 1 \end{matrix} \right]

\left[\begin{matrix}-\cfrac { 1 }{ \sqrt { 2 }}\\-\cfrac{1}{\sqrt{2}}\end{matrix} \right]

\left[\begin{matrix}\cfrac{ -1 }{\sqrt { 2 } }\\\cfrac{1}{\sqrt{ 2 } } \end{matrix} \right]

\left[ \begin{matrix} -1 \\ 0 \end{matrix} \right]