Let $\theta = \frac{\pi}{5}$ and $A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$. If $B = A + A^4$,then $\det(B)$

Let $A$ be a $3 \times 3$ matrix such that $A^2 - 5A + 7I = 0$.
Statement-$I$: ${A^{-1}} = \frac{1}{7}(5I - A)$.
Statement-$II$: The polynomial $A^3 - 2A^2 - 3A + I$ can be reduced to $5(A - 4I)$.

Let $\omega$ be the complex number $\cos \frac{2 \pi}{3} + i \sin \frac{2 \pi}{3}$. Then the number of distinct complex numbers $z$ satisfying $\left|\begin{array}{ccc} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{array}\right| = 0$ is equal to

Let $A = \begin{bmatrix} \frac{1}{6} & \frac{-1}{3} & \frac{-1}{6} \\ \frac{-1}{3} & \frac{2}{3} & \frac{1}{3} \\ \frac{-1}{6} & \frac{1}{3} & \frac{1}{6} \end{bmatrix}$. If $A^{2016l} + A^{2017m} + A^{2018n} = \frac{1}{\alpha} A$,for every $l, m, n \in N$,then the value of $\alpha$ is

The number of singular matrices of order $2 \times 2$,whose elements are from the set $\{2, 3, 6, 9\}$ is

If $A = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{bmatrix}$ and $\alpha, \beta, \gamma$ are the roots of the characteristic equation $|A - xI| = 0$,then $\alpha^2 + \beta^2 + \gamma^2 = $