1. ## determinat value

if A is symmetric matrix of odd order and aii=0 for all i then prove that determinant of A is an even number.

2. ## Re: determinat value

Has the matrix real entries?

3. ## Re: determinat value

yes, the entries are integers, other than diagonal elements.

4. ## Re: determinat value

We can use the definition of the determinant. Since $a_{ii}=0$, 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.