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

please explain.

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- May 13th 2012, 05:54 AMsaravananbsdeterminat value
if A is symmetric matrix of odd order and a

_{ii}=0 for all i then prove that determinant of A is an even number.

please explain. - May 15th 2012, 12:12 AMgirdavRe: determinat value
Has the matrix real entries?

- May 15th 2012, 09:53 AMsaravananbsRe: determinat value
yes, the entries are integers, other than diagonal elements.

- May 17th 2012, 02:16 AMgirdavRe: determinat value
We can use the definition of the determinant. Since $\displaystyle a_{ii}=0$, 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.