Matrices & Determinants
System of linear equations
Grade Class 12
Question:
Consider the matrices: A = \begin{pmatrix} 2 & -5 \\ 3 & m \end{pmatrix}, B = \begin{pmatrix} 20 \\ m \end{pmatrix} and X = \begin{pmatrix} x \\ y \end{pmatrix}. 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 ________.
Step-by-Step Solution
Key Concept: For a system AX=B, if det(A) != 0, the solution is X = A^-1 B. Calculate x and y in terms of m, apply the conditions x < 0 and y < 0 to find the interval (a, b), then compute the integral of the determinant of A.
The determinant |A| = 2m + 15. For a unique solution, 2m + 15 != 0. Using Cramer's rule, x = |A_x|/|A| = (20m + 5m)/(2m + 15) = 25m/(2m + 15) and y = |A_y|/|A| = (2m - 60)/(2m + 15). For x < 0, 25m/(2m + 15) < 0 implies m is in (-7.5, 0). For y < 0, (2m - 60)/(2m + 15) < 0 implies m is in (-7.5, 30). The intersection is (-7.5, 0), so a = -7.5 and b = 0. The integral is 8 * integral from -7.5 to 0 of (2m + 15) dm = 8 * [m^2 + 15m] from -7.5 to 0 = 8 * (0 - (56.25 - 112.5)) = 8 * 56.25 = 450.
Correct Answer: 450