Let A = bmatrix 2 & 1 x & y bmatrix be a matrix with integer entries. If A satisfies the equation A^3 - 27A + 10I = O , where I is the 2 2 identity matrix and O is the 2 2 zero matrix, then the value of |x - y| is equal to :

Step-by-step solution

Let P(t) = t^3 - 27t + 10 . The matrix A satisfies P(A) = O . By trial, t = 5 is a root of P(t) since 5^3 - 27(5) + 10 = 125 - 135 + 10 = 0 . Factoring P(t) , we get P(t) = (t - 5)(t^2 + 5t - 2) . Since A is a 2 2 matrix and is not a scalar multiple of the identity matrix (because its top-right entry is 1 0 ), its minimal polynomial must have degree 2 . The minimal polynomial of A must divide P(t) . Since A has integer entries, its characteristic polynomial (which is also its minimal polynomial here) must be a moni