Let $A$ be a $3 \times 3$ matrix with real entries such that

$$ A=\left[\begin{array}{ccc} 4 & -1 & \cos x \\ -1 & 5 x & 25 \\ x^2+1 & 25 & 7 \end{array}\right] $$

For how many values of $x$, the matrix $A$ is symmetric?

Let

$$ A=\left\{x \in \mathbf{R} \left\lvert\,-31<\operatorname{det}\left[\begin{array}{cc} 3 x-1 & 2 \\ -2 & 5 \end{array}\right] \leq 29\right.\right\} $$

Which one of the following statements is TRUE?

Let $M$ be a $3 \times 3$ matrix with real entries such that

$$ \left\{\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \end{array}\right]: M\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]\right\}=\left\{\left[\begin{array}{l} x_1 \\ x_2 \\ x_3 \end{array}\right]: x_1+x_2=0=x_2+x_3\right\} $$

What is the value of the determinant of M ?

Let $A$ be the matrix $\left[\begin{array}{ccc}\cos \theta & 0 & -\sin \theta \\ 1 & 1 & 1 \\ \sin \theta & 0 & \cos \theta\end{array}\right]$. For any natural number $k$, the determinant of $A^k$ is