What functions have the property $f(AB)=f(A)f(B)$ where $A,B$ are matrices? There is the classic example $\det(AB)=\det(A)\det(B)$. I am looking for other examples of this property, it is best if f maps matrices to scalars but any example would be great.
Note: No need for trivial $f(A)=cA$ type functions; I am looking for something more exotic.
 A: This might not be exactly what you want, but I think in general the functions you're describing are homomorphisms between a matrix group and some other group.
If you aren't familiar with groups, it's probably sufficient here to know that a group is a set with an invertible binary operation and an identity element; invertible matrices with the usual matrix multiplication are therefore an example, and this group is usually called the general linear group.
For example, the determinant is a homomorphism from the general linear group to the non-zero real numbers. I'm not sure if I have any examples of interesting maps, but using this terminology might make it easier for you to find some!
A: For $f:M_n(\mathbb F)\to\mathbb K$ where $\mathbb F$ and $\mathbb K$ are some fields and $M_n(\mathbb F)\ne M_2(GF(2))$, it is multiplicative if and only if $f=g\circ\det$ for some multiplicative map $g:\mathbb F\to\mathbb K$. As the other answer here has mentioned, the main idea has been outlined in Jyrki Lahtonen's answer to another question. The key fact is that when $n\ne2$ or $\mathbb F\ne GF(2)$, the special linear group $SL_n(\mathbb F)$ is a commutator subgroup of $GL_n(\mathbb F)$, i.e. every matrix of determinant $1$ can be expressed as a product of the form $BCB^{-1}C^{-1}$ for some $B,C\in GL_n(\mathbb F)$.
For $f:M_2(GF(2))\to\mathbb K$, it is not hard to show that $f$ is multiplicative if and only if $f=0$, $f=1$ or $f$ has the following properties:

*

*$f(A)=0$ for all singular matrices $A$,

*$f\left(\left\{I_2,\pmatrix{0&1\\ 1&1},\pmatrix{1&1\\ 1&0}\right\}\right)=\{1\}$,

*$f\left(\left\{\pmatrix{1&0\\ 1&1},\pmatrix{1&1\\ 0&1},\pmatrix{0&1\\ 1&0}\right\}\right)=\{1\}$ or $\{-1\}$.

If $\operatorname{char}(\mathbb K)=2$, such an $f$ is also in the form of $g\circ\det$ where $g:\mathbb F\to\mathbb K$ is multiplicative.
When $f:M_n(\mathbb F)\to M_m(\mathbb K)$ is matrix-valued, I don't know its characterisation. One easy example is $f(A)=\operatorname{diag}(\phi_1(A),\ldots,\phi_n(A))$ where $\phi_1,\phi_2,\ldots\phi_m$ are $m$ multiplicative maps from $M_n(\mathbb F)$ to $\mathbb K$. Since some $\phi_i$s can be taken to be zero, $f(I_n)$ can be singular but nonzero. A special case of this example (up to conjugation) is $f(A)=\phi(A)P$, where $\phi:M_n(\mathbb F)\to\mathbb K$ is a scalar-valued multiplicative map and $P\in M_m(\mathbb K)$ is a constant idempotent matrix.
