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 $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order $2$. If the roots of the equation $|A-xI|=0$ are $-1$ and $3$,then the sum of the diagonal elements of the matrix $A^2$ is $..............$

Let $A, B, C, D$ be square real matrices such that $C^T = DAB$,$D^T = ABC$,and $S = ABCD$. Then $S^2$ is equal to:

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:

Let $A = \begin{bmatrix} 5 & \sin^2 \theta & \cos^2 \theta \\ -\sin^2 \theta & -5 & 1 \\ \cos^2 \theta & 1 & 5 \end{bmatrix}$. Then the maximum value of $\det(A)$ is

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, then $A^3 - 4A^2 - 6A$ is equal to: