Let $$A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$$. If the sum of the diagonal elements of $$A^{13}$$ is $$3^n$$, then $$n$$ is equal to ________.

If the system of equations

$$\begin{aligned} & 2 x+7 y+\lambda z=3 \\ & 3 x+2 y+5 z=4 \\ & x+\mu y+32 z=-1 \end{aligned}$$

has infinitely many solutions, then $$(\lambda-\mu)$$ is equal to ______ :

Let $$\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}$$. If $$x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)$$ for some $$x, y, z \in \mathbb{R}, x y z \neq 0$$, then $$6 \alpha+4 \beta+\gamma$$ is equal to _________.

Let $$A$$ be a $$2 \times 2$$ symmetric matrix such that $$A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$$ and the determinant of $$A$$ be 1 . If $$A^{-1}=\alpha A+\beta I$$, where $$I$$ is an identity matrix of order $$2 \times 2$$, then $$\alpha+\beta$$ equals _________.