The total number of matrices A = 0 2 y 1 2 x y - 1 2 x - y 1 , x , y ∈ R , x ≠ y for which A T A = 3 I 3 is:

Step-by-step solution

Given A T A = 3 I 3 0 2 x 2 x 2 y y - y 1 - 1 1 0 2 y 1 2 x y - 1 2 x - y 1 = 3 0 0 0 3 0 0 0 3 8 x 2 0 0 0 6 y 2 0 0 0 3 = 3 0 0 0 3 0 0 0 3 On comparing the two matrices, we get 8 x 2 = 3 and 6 y 2 = 3 ⇒ x 2 = 3 8 and y 2 = 1 2 ⇒ x = ± 3 8 and y = ± 1 2 For each of x   &   y we have two possible values, hence there are total 2 × 2 = 4 combinations of values are possible. Hence, total 4 matrices are possible.