Question

the proof that the number of a symmetric matric is 1/2(n^2+n)

Answer 1

1. nxn matrix with excluded main diagonal contains n^2 - n numbers. In symmetric matrix, numbers over main diagonal mirror numbers under main diagonal, so there are (n^2-n)/2 numbers over main diagonal. Numbers over main diagonal and diagonal numbers define a symmetric matrix. Their quantity is (n^2-n)/2 + n or (n^2+n)/2 after simplification.