Let A = pmatrix x & 1 -1 & 0 pmatrix , where x R . If the number of elements in the set S = n 1, 2, , 100 : A^n = I is 33 , then the value of x^2 + 14 is

Step-by-step solution

Let the order of matrix A be k . The condition A^n = I is satisfied if and only if n is a multiple of k . The number of such multiples in the set 1, 2, , 100 is given by 100 k . We are given that 100 k = 33 . The only integer k that satisfies this is k = 3 . Thus, A^3 = I and clearly A I . The characteristic equation of A is ^2 - tr (A) + (A) = 0 . Here, tr (A) = x and (A) = 0 - (-1) = 1 . So, the characteristic equation is ^2 - x + 1 = 0 . Since A^3 = I and A is a real matrix with A I , its eigenvalues must be the