For the system of linear equation<br>2x - y + 3z = 5<br>3x + 2y - z = 7<br>4x + 5y + αz = β<br>Which of the following is(are) CORRECT?<br>(A) The system has infinitely many solutions for α = -5 and β = 9<br>(B) The system has a unique solution for α ≠ -5 and β = 8<br>(C) The system has infinitely many solutions for α = -6 and β = 9<br>(D) The system is inconsistent for α = -5 and β = 8
(A) The system has infinitely many solutions for α = -5 and β = 9
(B) The system has a unique solution for α ≠ -5 and β = 8
(C) The system has infinitely many solutions for α = -6 and β = 9
(D) The system is inconsistent for α = -5 and β = 8
Step-by-Step Solution
Key Concept: The system of linear equations has a unique solution if the determinant of the coefficient matrix is non-zero. If the determinant is zero, the system may have infinitely many solutions or be inconsistent depending on the consistency of the augmented matrix.
The coefficient matrix is A = [[2, -1, 3], [3, 2, -1], [4, 5, α]]. The determinant |A| = 2(2α + 5) + 1(3α + 4) + 3(15 - 8) = 4α + 10 + 3α + 4 + 21 = 7α + 35. For a unique solution, |A| ≠ 0, so 7α + 35 ≠ 0, which means α ≠ -5. If α = -5, |A| = 0. The augmented matrix is [[2, -1, 3, 5], [3, 2, -1, 7], [4, 5, -5, β]]. Performing row operations: R2 -> 2R2 - 3R1 gives [0, 7, -11, -1], R3 -> R3 - 2R1 gives [0, 7, -11, β - 10]. For the system to be consistent, the last two rows must be proportional, so β - 10 = -1, which means β = 9. If β ≠ 9, the system is inconsistent. Thus, for α = -5 and β = 8, the system is inconsistent.
Correct Answer: D