The determinant of a $4 \times 4$ matrix $A$ is 3 . The value of the determinant of $2 A$ is

$\_\_\_\_$ . (answer in integer)

Let $L_1$ and $L_2$ be two languages over a finite alphabet, such that $L_1 \cap L_2$ and $L_2$ are regular languages.

Which of the following statements is/are always true?

Let $\Sigma=\{a, b, c\}$. For $x \in \Sigma^{\star}$, and $\alpha \in \Sigma$, let $\#_\alpha(x)$ denote the number of occurrences of a in $x$. Which one or more of the following option(s) define(s) regular language(s)?

Let $\Sigma=\{1,2,3,4\}$ For $x \in \Sigma^*$, let prod $(x)$ be the product of symbols in $x$ modulo 7 . We take $\operatorname{prod}(\varepsilon)=1$, where $\varepsilon$ is the null string.

For example, $\operatorname{prod}(124)=(1 \times 2 \times 4) \bmod 7=1$.

Define $L=\left\{x \in \Sigma^{\star} \mid \operatorname{prod}(x)=2\right\}$.

The number of states in a minimum state DFA for $L$ is _________ (Answer in integer)