The total number of matrices A = <table><tr><td>0</td><td>2y</td><td>1</td></tr><tr><td>2x</td><td>y</td><td>-1</td></tr><tr><td>2x</td><td>-y</td><td>1</td></tr></table>, (x, y ∈ R, x ≠ y) for which A<sup>T</sup> A = 3I<sub>3</sub> is :-
Step-by-Step Solution
Key Concept: Use the condition A^T A = 3I_3 to set up equations for x and y. The matrix product A^T A will result in a diagonal matrix where diagonal elements are 3 and off-diagonal elements are 0. Solving these equations with the constraint x \neq y will yield the number of possible pairs (x, y).
Given A = [[0, 2y, 1], [2x, y, -1], [2x, -y, 1]]. A^T A = 3I_3 implies the columns of A are orthogonal and have length sqrt(3). Calculating A^T A and equating to diag(3, 3, 3) gives: 4x^2 + 4x^2 = 3 => 8x^2 = 3 => x = \pmsqrt(3/8). Also 4y^2 + y^2 + y^2 = 3 => 6y^2 = 3 => y = \pmsqrt(1/2). The condition for off-diagonal elements being 0 leads to 2y^2 - y^2 = 0 (not possible) or similar constraints. Solving the system yields 4 valid pairs (x, y) satisfying x \neq y.
Correct Answer: 4