Let A = bmatrix a & b c & d bmatrix be a 2 2 matrix where the entries a, b, c, d are chosen independently and at random from the set 0, 1, 2 . Consider the system of homogeneous linear equations AX = O , where X = bmatrix x y bmatrix and O = bmatrix 0 0 bmatrix . The probability that this system of equations has a non-trivial solution is :

Step-by-step solution

A homogeneous system of linear equations AX = O has a non-trivial solution if and only if the determinant of the coefficient matrix A is zero. For A = bmatrix a & b c & d bmatrix , we require |A| = ad - bc = 0 , which means ad = bc . Since each entry is chosen from 0, 1, 2 , the total number of possible matrices is 3^4 = 81 . Let us find the number of ordered pairs (u, v) from 0, 1, 2 0, 1, 2 that yield each possible product uv : Product = 0 : (0,0), (0,1), (1,0), (0,2), (2,0) 5 pairs. Product = 1 : (1,1) 1 pair. P