Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers. Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$. What is the value of $c$? (Single choice question, 3 points) (1) 0 (2) $- 1$ (3) 1 (4) $- \frac { 1 } { 2 }$ (5) $\frac { 1 } { 2 }$
Let $A = \left[ \begin{array} { l l } a _ { 1 } & a _ { 2 } \\ a _ { 3 } & a _ { 4 } \end{array} \right]$ and $B = \left[ \begin{array} { l l } b _ { 1 } & b _ { 2 } \\ b _ { 3 } & b _ { 4 } \end{array} \right]$ both be rotation matrices on the coordinate plane with center at the origin $O$, rotating counterclockwise by an acute angle, and satisfying $A ^ { 2 } = B ^ { 3 } = \left[ \begin{array} { l l } 0 & c \\ 1 & d \end{array} \right]$, where $c$ and $d$ are real numbers.
Let point $P ( 1,1 )$ be transformed by $A ^ { 3 }$ to point $Q$, and point $Q$ be transformed by $B ^ { 4 }$ to point $R$.
What is the value of $c$? (Single choice question, 3 points)\\
(1) 0\\
(2) $- 1$\\
(3) 1\\
(4) $- \frac { 1 } { 2 }$\\
(5) $\frac { 1 } { 2 }$