Matrices & Determinants
System of linear equations
Grade Class 12
Question:
Let S<sub>1</sub> and S<sub>2</sub> be respectively the sets of all a ∈ R - {0} for which the system of linear equations <br> ax + 2ay - 3az = 1 <br> (2a + 1)x + (2a + 3)y + (a + 1)z = 2 <br> (3a + 5)x + (a + 5)y + (a + 2)z = 3 <br> has unique solution and infinitely many solutions. Then
(1) n(S<sub>1</sub>) = 2 and S<sub>2</sub> is an infinite set
(2) S<sub>1</sub> is an infinite set n(S<sub>2</sub>) = 2
(3) S<sub>1</sub> = φ and S<sub>2</sub> = R - {0}
(4) S<sub>1</sub> = R - {0} and S<sub>2</sub> = φ
Step-by-Step Solution
Key Concept: A system of linear equations has a unique solution if the determinant of the coefficient matrix is non-zero. It has infinitely many solutions if the determinant is zero and the augmented matrix satisfies specific consistency conditions.
The determinant of the coefficient matrix D = |a 2a -3a; 2a+1 2a+3 a+1; 3a+5 a+5 a+2|. Calculating this determinant, we find it is non-zero for all a \in R - {0}. Thus, the system always has a unique solution for all a \in R - {0}, meaning S1 = R - {0} and S2 = \phi.
Correct Answer: (4)