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 then performing operations, we find that 1/a, 1/b, 1/c are in A.P.
Correct Answer: (4)