Let A be a $$3 \times 3$$ matrix and $$\operatorname{det}(A)=2$$. If $$n=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots . .(\operatorname{adj} A))}_{2024-\text { times }}))$$, then the remainder when $$n$$ is divided by 9 is equal to __________.

Let for any three distinct consecutive terms $$a, b, c$$ of an A.P, the lines $$a x+b y+c=0$$ be concurrent at the point $$P$$ and $$Q(\alpha, \beta)$$ be a point such that the system of equations

$$\begin{aligned} & x+y+z=6, \\ & 2 x+5 y+\alpha z=\beta \text { and } \end{aligned}$$

$$x+2 y+3 z=4$$, has infinitely many solutions. Then $$(P Q)^2$$ is equal to _________.

Let $$A$$ be a $$2 \times 2$$ real matrix and $$I$$ be the identity matrix of order 2. If the roots of the equation $$|\mathrm{A}-x \mathrm{I}|=0$$ be $$-1$$ and 3, then the sum of the diagonal elements of the matrix $$\mathrm{A}^2$$ is