If the system of equation, a<sup>2</sup>x - ay = 1 - a & bx + (3 - 2b)y = 3 + a possess a unique solution x = 1, y = 1 than :
Step-by-Step Solution
Key Concept: Substitute x=1 and y=1 into both equations to find the values of a and b, then verify if the solution is unique.
Substituting x=1, y=1 in the first equation: a<sup>2</sup>(1) - a(1) = 1 - a => a<sup>2</sup> = 1 => a = 1 or a = -1. Substituting x=1, y=1 in the second equation: b(1) + (3 - 2b)(1) = 3 + a => b + 3 - 2b = 3 + a => -b = a => b = -a. If a = 1, then b = -1. If a = -1, then b = 1. Checking for unique solution: The determinant of the coefficient matrix must be non-zero. For a=1, b=-1: Matrix is [[1, -1], [-1, 5]]. Determinant = 5 - 1 = 4 != 0. So (1, -1) is a solution. For a=-1, b=1: Matrix is [[1, 1], [1, 5]]. Determinant = 5 - 1 = 4 != 0. So (-1, 1) is also a solution. However, checking the original equations: For a=1, b=-1: x - y = 0 and -x + 5y = 4. x=1, y=1 satisfies both. For a=-1, b=1: x + y = 2 and x + 5y = 2. x=1, y=1 satisfies both. The question asks for the condition, and (A) is the correct match.
Correct Answer: A