Matrices & Determinants
System of Linear Equations
Grade Class 12
Question:
Let a, λ, μ ∈ R. Consider the system of linear equations<br>ax + 2y = λ<br>3x - 2y = μ<br>Which of the following statement(s) is(are) correct?
(A) if a = -3, then the system has infinitely many solutions for all values of λ and μ
(B) if a ≠ -3, then the system has a a unique solution for all values of λ and μ
(C) if λ + μ = 0, then the system has infinitely many solutions for a = -3
(D) if λ + μ ≠ 0, then the system has no solution for a = -3
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 no solution depending on the consistency of the equations.
The system is:<br>ax + 2y = \lambda<br>3x - 2y = \mu<br>The determinant of the coefficient matrix is D = |a 2; 3 -2| = -2a - 6 = -2(a + 3).<br>For a unique solution, D \neq 0, which means a \neq -3. Thus, (B) is correct.<br>If a = -3, the system becomes:<br>-3x + 2y = \lambda<br>3x - 2y = \mu<br>Adding the two equations gives 0 = \lambda + \mu.<br>If \lambda + \mu = 0, the equations are consistent and dependent, leading to infinitely many solutions. Thus, (C) is correct.<br>If \lambda + \mu \neq 0, the equations are inconsistent, leading to no solution. Thus, (D) is correct.
Correct Answer: B,C,D