S. Carter proved that every $\mathbb F$ valued map $f$ on square matrices which satisfies $f(ABC)=f(CBA)$ can be written as $f(X)=\pi(\det(X))$ for a unique map $\pi:\mathbb F\to \mathbb F$. Also $f$ is multiplicative iff $\pi$ is multiplicative.

When you assume $\mathbb F=\mathbb R$ and $f$ is continuous, for example, then a continuous multiplicative $\pi$ is of the form $x^{r}$ for some $r$. And the result you quote is a corollary. This general result is proved in "Scalar valued mappings of squared matrices".

S. Cater proved that every $\mathbb F$ valued map $f$ on square matrices which satisfies $f(ABC)=f(CBA)$ can be written as $f(X)=\pi(\det(X))$ for a unique map $\pi:\mathbb F\to \mathbb F$. Also $f$ is multiplicative iff $\pi$ is multiplicative.

When you assume $\mathbb F=\mathbb R$ and $f$ is continuous, for example, then a continuous multiplicative $\pi$ is of the form $x^{r}$ for some $r$. And the result you quote is a corollary,(because polynomials are continuous). This general result is proved in "Scalar valued mappings of squared matrices".