If $A = \begin{bmatrix} p & q & r \\ r & p & q \\ q & r & p \end{bmatrix}$ and $A A^T = I$,then $p^3 + q^3 + r^3 =$ . . . . . .

If $\frac{x^2+7}{(x^2+1)(x-2)}=\frac{A}{x-2}+\frac{Bx+C}{x^2+1}$, then the determinant of the matrix $\begin{bmatrix} A & B \\ C & \frac{2}{5} \end{bmatrix}$ is

If $P$ and $Q$ are square matrices such that $P^{2006} = O$ and $PQ = P + Q$,then $\det(Q)$ will be

In a $\Delta ABC,$ if $\left| \begin{array}{ccc} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{array} \right| = 0$,then $\sin^2 A + \sin^2 B + \sin^2 C = $

Let the minimum $m$ $(m \in Z^+)$ be defined as the power of a square matrix $A$ such that $A^m = I$. If $A^5 = I$ and $ABA^{-1} = B^2$,then the power of matrix $B$ such that $B^k = I$ is between:

If $A=\begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix}$,$B=\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$,$C=ABA^T$ and $X=A^T C^2 A$,then $\operatorname{det}(X)$ is equal to: