if A is symmetric matrix of odd order and aii=0 for all i then prove that determinant of A is an even number.
We can use the definition of the determinant. Since , we can sum over the permutations without fixed point. Since these permutations act over a set which have an odd number of elements, these one have a square different from the identity. So you can write the sum as two equal parts, which will give an even number.