Consider the matrices : $$A = \begin{bmatrix} 2 & -5 \\ 3 & m \end{bmatrix}, B = \begin{bmatrix} 20 \\ m \end{bmatrix}$$ and $$X = \begin{bmatrix} x \\ y \end{bmatrix}$$. Let the set of all $$m$$, for which the system of equations $$AX = B$$ has a negative solution (i.e., $$x < 0$$ and $$y < 0$$), be the interval $$(a, b)$$. Then $$8\int_a^b |A| \, dm$$ is equal to ________

System equations:

$$2x - 5y = 20$$

$$3x + my = m$$

Using Cramer's Rule:

$$\Delta = |A| = 2m + 15, \quad \Delta_x = 20m + 5m = 25m, \quad \Delta_y = 2m - 60$$

For negative solutions ($$x < 0, y < 0$$): * Case $$\Delta > 0 \implies m > -7.5$$: Requires $$25m < 0 \implies m < 0$$ and $$2m - 60 < 0 \implies m < 30$$. Intersection: $$(-7.5, 0)$$.

Case $$\Delta < 0 \implies m < -7.5$$: Requires $$25m > 0$$ (Impossible for $$m < -7.5$$).

Interval: $$(a,b) = (-7.5, 0) \implies a = -7.5, b = 0$$.

Integral: $$8 \int_{-7.5}^{0} (2m + 15) \, dm = 8 \left[ m^2 + 15m \right]_{-7.5}^{0} = 8 \left(0 - (56.25 - 112.5)\right) = 8 \times 56.25 = \mathbf{450}$$