Let A = 1, 2, 3, , 100 . A relation R on the set A is represented by a 100 100 matrix M = [m_ ij ] , where m_ ij = 1 if i is a prime number of the form 4k+1 (for some integer k ) and j is a perfect square, and m_ ij = 0 otherwise. The sum of all elements of the matrix M is

Step-by-step solution

The sum of all elements of the incidence matrix M is equal to the number of ordered pairs (i, j) in the relation R . The condition for m_ ij = 1 is that i is a prime of the form 4k+1 and j is a perfect square, with both i, j 1, 2, , 100 . First, we count the number of perfect squares up to 100 . These are 1^2, 2^2, , 10^2 , which gives 10 possible values for j . Next, we count the number of prime numbers up to 100 that are of the form 4k+1 . These primes are: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97 . Counting the