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 $A = \int_{1}^{\sin \theta} \frac{t}{1+t^2} dt$ 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| = $

For each real number $x$ such that $-1 < x < 1$,let $A(x)$ be the matrix $\frac{1}{1-x^2} \begin{bmatrix} 1 & -x \\ -x & 1 \end{bmatrix}$. If $z = \frac{x+y}{1+xy}$,then which of the following is true?

The number of elements in the set $\{A=\begin{bmatrix} a & b \\ 0 & d \end{bmatrix} : a, b, d \in \{-1, 0, 1\} \text{ and } (I-A)^3 = I-A^3 \}$,where $I$ is the $2 \times 2$ identity matrix,is:

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

For some $\alpha, \beta \in R$, let $A = \begin{bmatrix} \alpha & 2 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 1 \\ 1 & \beta \end{bmatrix}$ be such that $A^{2} - 4A + 2I = B^{2} - 3B + I = O$. Then $(\text{det}(\text{adj}(A^{3} - B^{3})))^{2}$ is equal to ....