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 an 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: For a system of linear equations AX = B, the system has a unique solution if det(A) \neq 0 and infinitely many solutions if det(A) = 0 and the augmented matrix [A|B] has rank less than the number of variables.
The determinant of the coefficient matrix A is D = |a 2a -3a; 2a+1 2a+3 a+1; 3a+5 a+5 a+2|. Calculating this determinant, we find D = a( (2a+3)(a+2) - (a+5)(a+1) ) - 2a( (2a+1)(a+2) - (3a+5)(a+1) ) - 3a( (2a+1)(a+5) - (3a+5)(2a+3) ). Simplifying this, D = a( a^2 + 4a + 1 ) - 2a( -a^2 - 4a - 3 ) - 3a( -4a^2 - 19a - 10 ). Further simplification shows D is never zero for a \neq 0. Thus, the system always has a unique solution for all a \in R - {0}. Therefore, S1 = R - {0} and S2 = \phi.
Correct Answer: 4