Consider the matrices : $$A=\left[\begin{array}{cc}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$$ and $$X=\left[\begin{array}{l}x \\ y\end{array}\right]$$. Let the set of all $$m$$, for which the system of equations $$A X=B$$ has a negative solution (i.e., $$x<0$$ and $$y<0$$), be the interval $$(a, b)$$. Then $$8 \int_\limits a^b|A| d m$$ is equal to _________.

Let $$A$$ be a non-singular matrix of order 3. If $$\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}$$ and $$\operatorname{det}(\operatorname{adj}(2 \mathrm{~A}))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}$$, then $$|3 \mathrm{~m}+2 \mathrm{n}|$$ is equal to _________.

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 ______ :