Show that if a matrix $A$ satisfies $A=-A^T$, then all the diagonal elements of the matrix are $0$.

Question

Show that if a matrix $A$ satisfies $A=-A^T$, then all the diagonal elements of the matrix are $0$.

I can show that $A+A^T=0$ and from there I can see that all the elements would cancel eachother out. But how does this imply that the tr$(A)$ would be $0$ as well?

Answer 1

Since transpose operation leaves diagonal of $A$ intact. If diagonal of $A$ is non-zero then the sum $A + A^T$ will have non-zero diagonal as well. Therefore diagonal must be zero.

Answer 2

If the matrix is given over a field with $\mathrm{char}(K)\neq 2$, then due to there being no zero divisors, entries on the diagonal must be $0$.