If $A,B,C$are three $n \times n$ matrices, then $(ABC)' = $

If $ A$  is a symmetric matrix and $n \in N$, then ${A^n}$ is

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\left[\begin{array}{lll}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]$ where each of $a, b$, and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

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

If $A=\left[\begin{array}{lll}3 & \sqrt{3} & 2 \\ 4 & 2 & 0\end{array}\right]$ and $B=\left[\begin{array}{rrr}2 & -1 & 2 \\ 1 & 2 & 4\end{array}\right],$ verify that $(A+B)^{\prime}=A^{\prime}+B^{\prime}$.