Inverse of Hodge star operator

Question

According to Wikipedia [https://en.wikipedia.org/wiki/Hodge_star_operator] the inverse of Hodge star operator $*:\Lambda^k\rightarrow \Lambda^{n-k}$ is $*^{-1}:\Lambda^k\rightarrow \Lambda^{n-k}$ defined by
$$\eta\rightarrow (-1)^{k(n-k)}*\eta.$$
But why it is defined in that way? How is it the inverse of $*$ operator? The definition seems it has been deduced from the twice Hodge star operator but how does it work?

Answer 1

An example: if $v = e^1\wedge e^2 \wedge e^3 \wedge e^4$ is the volume form using an orthonormal basis, then since $(e^1) \wedge (e^2\wedge e^3\wedge e^4)= v$, $$\star (e^1) = e^2\wedge e^3 \wedge e^4$$ and since $(e^2\wedge e^3\wedge e^4)\wedge (e^1)= -e^1\wedge e^2 \wedge e^3 \wedge e^4 = -v$, $$\star(e^2\wedge e^3\wedge e^4) = -e^1.$$

But as another user said, you should compute $\star \star$ on a general (orthonormal basis) element, and see basically the above happens.