Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A 2 is 1 , then the possible number of such matrices is:

Step-by-step solution

A = a b b c ,    a , b , c ∈ I A 2 = a b b c a b b c = a 2 + b 2 b a + c b a + c b 2 + c 2 Sum of the diagonal entries of A 2 = a 2 + 2 b 2 + c 2 Given a 2 + 2 b 2 + c 2 = 1 , a , b , c ∈ I b = 0   &   a 2 + c 2 = 1 Case-1: a = 0 ⇒ c = ± 1 ( 2 -matrices) Case-2: c = 0 ⇒ a = ± 1 ( 2 -matrices) Total = 4 matrices