If $C$ and $D$ are two $n \times n$ non-singular matrices over the set of real numbers $\mathbb{R}$ such that $CD = -DC$, then $n$ is:

Let $X = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$. Let $Y$ be a $2 \times 2$ real matrix satisfying the condition $XY = YX$. Then the smallest possible value of $\det(Y)$ is

If the elements of matrix $A$ are the reciprocals of elements of matrix $\left[\begin{array}{ccc}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right]$,where $\omega$ is a complex cube root of unity,then:

For some $\alpha, \beta \in R$, let $A = \begin{bmatrix} \alpha & 2 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 1 \\ 1 & \beta \end{bmatrix}$ be such that $A^{2} - 4A + 2I = B^{2} - 3B + I = O$. Then $(\text{det}(\text{adj}(A^{3} - B^{3})))^{2}$ is equal to ....

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

Let $A, B$ be two $3 \times 3$ matrices and $C$ be a $3 \times 3$ identity matrix such that $AB-C$ is a non-singular matrix. Let $D=(AB-C)^{-1}$. Then, consider the following statements.
Statement $I$: $\operatorname{det}(BA)=\operatorname{det}(BA-C) \operatorname{det}(BDA)$
Statement $II$: $ABD=DAB$
Which of the above statements is (are) true?