Matrices & Determinants
System of linear equations
Grade Class 12
Question:
If the system of linear equations <br> 2x + 2ay + az = 0 <br> 2x + 3by + bz = 0 <br> 2x + 4cy + cz = 0 <br> where a, b, c ∈ R are non-zero and distinct; has a non-zero solution, then :
(1) a, b, c are in A.P.
(2) a + b + c = 0
(3) a, b, c are in G.P.
(4) 1/a, 1/b, 1/c are in A.P.
Step-by-Step Solution
Key Concept: For a homogeneous system of linear equations to have a non-zero solution, the determinant of the coefficient matrix must be zero.
The system is homogeneous. For a non-zero solution, the determinant of the coefficient matrix must be zero: <br> | 2 2a a | <br> | 2 3b b | = 0 <br> | 2 4c c | <br> Taking 2 common from C1 and a, b, c common from C2 and C3 respectively: <br> 2abc | 1/a 2 1 | <br> | 1/b 3 1 | = 0 <br> | 1/c 4 1 | <br> Since a, b, c are non-zero, the determinant must be zero. Performing R2 → R2 - R1 and R3 → R3 - R1: <br> | 1/a 2 1 | <br> | 1/b - 1/a 1 0 | = 0 <br> | 1/c - 1/a 2 0 | <br> Expanding along C3: 1 * (2(1/b - 1/a) - 1(1/c - 1/a)) = 0 <br> 2/b - 2/a - 1/c + 1/a = 0 <br> 2/b - 1/a - 1/c = 0 <br> 2/b = 1/a + 1/c <br> This implies 1/a, 1/b, 1/c are in A.P.
Correct Answer: 4