$A$ determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability that the value of the chosen determinant is positive is:

Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A^2 = 3A + \alpha I$. If $A^4 = 21A + \beta I$,then:

Let $ABC = I$. Then $tr(ABC + BCA + CAB)$ is (where the order of matrices $A, B, C$ is $3 \times 3$ and $tr(A)$ is the sum of the diagonal elements in $A$).

Let $M$ and $N$ be two $3 \times 3$ matrices such that $MN = NM$. Further,if $M \neq N^2$ and $M^2 = N^4$,then:
$(A)$ determinant of $(M^2 + MN^2)$ is $0$
$(B)$ there is a $3 \times 3$ non-zero matrix $U$ such that $(M^2 + MN^2)U$ is the zero matrix
$(C)$ determinant of $(M^2 + MN^2) \geq 1$
$(D)$ for a $3 \times 3$ matrix $U$,if $(M^2 + MN^2)U$ equals the zero matrix then $U$ is the zero matrix

If $A = \begin{bmatrix} 0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0 \end{bmatrix}$ and $I$ is the identity matrix of order $2$,show that $I+A = (I-A) \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$.

If $\alpha = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3}$,then the value of the determinant $\left| \begin{array}{ccc} 1 & \alpha & \alpha^2 \\ \alpha^2 & 1 & \alpha \\ \alpha & \alpha^2 & 1 \end{array} \right|$ is