Multiplicative functions on matrices What are some examples of multiplicative functions on matrices?
More precisely, I'm looking for $f : M^{n \times n}, M^{n \times n} \to \mathbb R$ with the property $$f(AB) = f(A)f(B)$$ where A, B are $n \times n$ matrices. A particularly well-known example is the determinant, but I'm curious about other examples, and possibly a classification of them.
 A: I assume $R$ is the reals, and these are matrices over the reals.
Of course there are the trivial examples $f(A) = 0$ for all $A$ and $f(A) = 1$ for all $A$.  Any nontrivial example will have $f(I) = 1$, $f(0) = 0$, and $f(A^{-1}) = f(A)^{-1}$ for invertible $A$.  Thus $f$ is invariant under similarity.  
If $A$ is singular, there are invertible matrices $S_1, S_2, \ldots, S_{n-1}$ such that $A S_{1} A \ldots S_{n-1} A = 0$, and therefore (if $f$ is nontrivial) $f(A) = 0$.
You might try $f(A) = |\det(A)|^r$ or $f(A) = \text{sgn}(\det(A)) |\det(A)|^r$
for invertible $A$, where $r$ is any real constant, with $f(A) = 0$ if $\det A = 0$.
EDIT: 
These are all measurable solutions.  In fact, 
 $SL(n,R)$ is the commutator subgroup of $GL(n,R)$, and since $f$ is $1$ on any commutator $ABA^{-1}B^{-1}$, $f(A) = 1$ whenever $\det(A) = 1$.
Let $C$ be the diagonal matrix with diagonal entries $(-1,1,\ldots, 1)$ (so that $\det(C) = -1$ and $C^2 = I$).
  Any nonsingular matrix $A$ can then be written as $A = t B$ (if $\det A > 0$) or $A = t B C$ (if $\det A < 0$), where $t = 
 |\det(A)|^{1/n}$ and $\det(B) = 1$.  We then have $f(A) = f(tI)$ or $-f(tI)$.
$f(tI)$ is a multiplicative function on the positive reals, and the only such functions that are measurable are $t \to t^r$ for real constants $r$.
