Question

Consider the matrices: $A = \begin{bmatrix} 2 & -5 \\\ 3 & m \end{bmatrix}, \quad B = \begin{bmatrix} 20 \\\ m \end{bmatrix}, \quad \text{and} \quad X = \begin{bmatrix} x \\\ y \end{bmatrix}.$ 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 $8 \int_a^b |\det(A)| \, dm$ is equal to _____ .

Answer 1

Given:

$A = \begin{pmatrix} 2 & -5 \\\ 3 & m \end{pmatrix}, \quad B = \begin{pmatrix} 20 \\\ m \end{pmatrix}, \quad X = \begin{pmatrix} x \\\ y \end{pmatrix}$

From the equations:

$2x - 5y = 20$ (1)

$3x + my = m$ (2)

We get:

$y = \frac{2m - 60}{2m + 15}$

For $y < 0$, $m \in \left(-\frac{15}{2}, 30\right)$.

Similarly:

$x = \frac{25m}{2m + 15}$

For $x < 0$, $m \in \left(-\frac{15}{2}, 0\right)$.

Thus, combining conditions:

$m \in \left(-\frac{15}{2}, 0\right)$

The determinant of matrix $A$ is:

$|A| = 2m + 15$

Now:

$8 \int_{-\frac{15}{2}}^{0} (2m + 15) \, dm = 8 \left[ m^2 + 15m \right]_{-\frac{15}{2}}^{0}$

= 8 \left\\{ \frac{225}{4} - \frac{225}{2} \right\\}

$= 8 \times \frac{225}{4} = 450$

Final Answer: 450