If $a_{r}=(\cos 2 r \pi+i \sin 2 r \pi)^{1 / 9}$,then the value of $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to

The number of integers $x$ satisfying $-3 x^4 + \operatorname{det}\begin{bmatrix} 1 & x & x^2 \\ 1 & x^2 & x^4 \\ 1 & x^3 & x^6 \end{bmatrix} = 0$ is equal to

For $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 1 & 3 \\ 1 & 1 & 0 \end{bmatrix}$,if $A^3 - 2A^2 + kA - 4I_3 = 0$,then $k = $ . . . . . . .

Let $A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$ and $P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}$. The sum of the prime factors of $|P^{-1}AP - 2I|$ is equal to

If $\omega$ is an imaginary cube root of unity,then the value of $\left[\begin{array}{ccc}1 & \omega^{2} & 1-\omega^{4} \\ \omega & 1 & 1+\omega^{5} \\ 1 & \omega & \omega^{2}\end{array}\right]$ is

If $A = \int\limits_1^{\sin \theta } {\frac{t}{{1 + {t^2}}}} dt$ and $B = \int\limits_1^{\csc \theta } {\frac{dt}{{t\left( {1 + {t^2}} \right)}}} $,(where $\theta \in \left( {0, \frac{\pi }{2}} \right)$),then the value of $\left| {\begin{array}{*{20}{c}} A & {{A^2}} & { - B} \\ {{e^{A + B}}} & {{B^2}} & { - 1} \\ 1 & {{A^2} + {B^2}} & { - 1} \end{array}} \right|$ is