if A is symmetric matrix of odd order and aii=0 for all i then prove that determinant of A is an even number.
please explain.
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if A is symmetric matrix of odd order and aii=0 for all i then prove that determinant of A is an even number.
please explain.
Has the matrix real entries?
yes, the entries are integers, other than diagonal elements.
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.