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 \(P^3X\).

\[\left[ \begin{matrix} 0 \\ 1 \end{matrix} \right] \] \[\left[ \begin{matrix} -1 \\ 0 \end{matrix} \right] \] \[\left[\begin{matrix}\cfrac{ -1 }{\sqrt { 2 } }\\\cfrac{1}{\sqrt{ 2 } } \end{matrix} \right] \] \[\left[\begin{matrix}-\cfrac { 1 }{ \sqrt { 2 }}\\-\cfrac{1}{\sqrt{2}}\end{matrix} \right]\]

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