If $A$ is a non-singular matrix such that $(A-2 I)$ $(A-3 I)=0$, then $\frac{1}{5} A+\frac{6}{5} A^{-1}=$

Let $A$ be a matrix such that $A B$ is a scalar matrix, where $B=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]$ and $\operatorname{det}(3 A)=27$. Then, $3 A^{-1}+A^2=$

If $A$ is a symmetric matrix with real entries, then

$$ \begin{aligned} &\text { If } \omega \neq 1 \text { is a cube root of unity, then }\\ &\left|\begin{array}{ccc} \omega+\omega^2 & \omega^2+\omega^9 & \omega^9+\omega \\ \omega^{27}+\omega^{31} & \omega^{31}+\omega^{17} & \omega^{17}+\omega^{27} \\ \omega^{30}+\omega^{41} & \omega^{41}+\omega^{19} & \omega^{19}+\omega^{30} \end{array}\right|= \end{aligned} $$