here is the quastion:
Let n be an odd integer and let A = (aij) be an n × n matrix that satisfies
the condition aij + aji = 0 for all 1 =< (i, j )=< n. Prove that |A| = 0.
i tried every possible way which i know but didnt succeed
tanq
you always should mention which field (or ring) the entries of your matrix come from. for example the cliam in your problem would be false if the entries of A were in a domain of characteristic 2.
otherwise the claim is true and very easy to prove: since $\displaystyle A=-A^T,$ we have: $\displaystyle |A|=|-A^T|=(-1)^n|A^T|=(-1)^n|A|=-|A|,$ because $\displaystyle n$ is odd. therefore $\displaystyle 2|A|=0$ and hence $\displaystyle |A|=0.$