Prove $I(Z,X) = -I(X,Z)$.

Question

I want to show that $I(Z,X) = -I(X,Z)$.

So clearly I have two orientations for $X$ and $Z$ each. Do I discuss these four cases, each consider $I(Z,X)$ and $I(X,Z)$? Is this the correct approach and is there a less brute-force way to do so?

Thank you~

Answer 1

In general, if $\dim X=k$ and $\dim Z=\ell$, then $I(Z,X)=(-1)^{k\ell} I(X,Z)$. Use the definition with ordered bases.