Let $A$ be a symmetric matrix and $B$ be a skew-symmetric matrix,such that $A - B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Then $|A|$ is:

If the matrix $A = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$ is expressed as the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, then which of the following relations is correct?

If $S = \{x \in [0, 2\pi] : \begin{vmatrix} 0 & \cos x & -\sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{vmatrix} = 0\}$,then $\sum_{x \in S} \tan \left( \frac{\pi}{3} + x \right)$ is equal to

If $A = \begin{bmatrix} \cos^2 \alpha & \cos \alpha \sin \alpha \\ \cos \alpha \sin \alpha & \sin^2 \alpha \end{bmatrix}$ and $B = \begin{bmatrix} \cos^2 \beta & \cos \beta \sin \beta \\ \cos \beta \sin \beta & \sin^2 \beta \end{bmatrix}$ are two matrices such that the product $AB$ is a null matrix,then $\alpha - \beta$ is:

If $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$,prove that $A^{3} - 6A^{2} + 7A + 2I = 0$.

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: