Question:

Which of the following options is/are correct?<br>(A) Let $A = \begin{bmatrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \end{bmatrix}$ are two matrices such that $AB = (AB)^{-1}$ and $AB \neq I$ (where $I$ is an identity matrix of order $3 \times 3$). Then $tr\left(AB + (AB)^2 + (AB)^3 + ..... + (AB)^{100}\right)$ equals 100.<br>(B) If $A$ and $B$ are square matrices of order 3, where $|A| = -2$ and $|B| = 1$, then $|(A^{-1})adj(B^{-1})adj(2A^{-1})|$ equals 8.<br>(C) If $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then $F(x).F(y) = F(x + y)$ and $[F(x)]^{-1} = F(-x)$.<br>(D) Let $X$ be the solution set of the equation $A^x = I$, where $A = \begin{bmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}$ and $I$ is the corresponding unit matrix and $X \subseteq \mathbb{N}$ then the minimum value of $\sum(\cos^n \theta + \sin^n \theta), \theta \in \mathbb{R}, n \in \mathbb{Z}$ is 2.