A 2 2 matrix A is defined as A = bmatrix a & b c & 2a bmatrix , where the elements a, b, c are chosen independently with replacement from the set 1, 2, 3, , 10 . The probability that the homogeneous system of linear equations A X = 0 has infinitely many solutions is :

Step-by-step solution

The total number of possible matrices A is 10 10 10 = 1000 . For the homogeneous system A X = 0 to have infinitely many solutions, the determinant of matrix A must be zero: (A) = a(2a) - bc = 0 2a^2 = bc We need to find the number of ordered triplets (a, b, c) from the given set that satisfy this condition: For a = 1 , bc = 2 (b, c) = (1, 2), (2, 1) (2 cases). For a = 2 , bc = 8 (b, c) = (1, 8), (2, 4), (4, 2), (8, 1) (4 cases). For a = 3 , bc = 18 (b, c) = (2, 9), (3, 6), (6, 3), (9, 2) (4 cases). For a = 4 , bc =