Matrices & Determinants
System of linear equations
Grade Class 12
Question:
System of equation x + y + az = b, 2x + 3y = 2a & 3x + 4y + a^2z = ab + 2 has
(A) unique solution when a ≠ 0, b ∈ R
(B) no solution when a = 0, b = 1
(C) infinite solution when a = 0, b = 2
(D) infinite solution when a = 1, b ∈ R
Step-by-Step Solution
Key Concept: Analyze the determinant of the coefficient matrix and use Cramer's rule or Gaussian elimination to determine the nature of solutions based on the values of a and b.
The coefficient matrix is A = [[1, 1, a], [2, 3, 0], [3, 4, a^2]]. The determinant |A| = 1(3a^2 - 0) - 1(2a^2 - 0) + a(8 - 9) = 3a^2 - 2a^2 - a = a^2 - a = a(a - 1). For a unique solution, |A| != 0, so a != 0 and a != 1. Checking the options, (A) is correct.
Correct Answer: 1