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 $a, b, c, d, e, f$ are in $G.P.$,then the value of $\left| \begin{array}{ccc} a^2 & d^2 & x \\ b^2 & e^2 & y \\ c^2 & f^2 & z \end{array} \right|$ depends on

Let $A$ and $B$ be real matrices of the form $\begin{bmatrix} \alpha & 0 \\ 0 & \beta \end{bmatrix}$ and $\begin{bmatrix} 0 & \gamma \\ \delta & 0 \end{bmatrix}$,respectively.
Statement $1$: $AB - BA$ is always an invertible matrix.
Statement $2$: $AB - BA$ is never an identity matrix.

Let $[A]_{3 \times 3}$ be a non-singular matrix such that $A^{-1}=\frac{1}{3}(A^2-5A+7I)$. Then $17A^8-85A^7+119A^6-51A^5-19A^4+95A^3-133A^2+58A+I=$

If $A, B, C$ are the angles of a triangle,then the value of the determinant $\left| \begin{array}{ccc} \sin 2A & \sin C & \sin B \\ \sin C & \sin 2B & \sin A \\ \sin B & \sin A & \sin 2C \end{array} \right|$ is

If $A = \int_{1}^{\sin \theta} \frac{t}{1+t^2} dt$ and $B = \int_{1}^{\operatorname{cosec} \theta} \frac{1}{t(1+t^2)} dt$,then the value of $\left| \begin{array}{ccc} A & A^2 & B \\ e^{A+B} & B^2 & -1 \\ 1 & A^2+B^2 & -1 \end{array} \right| = $