To be honest, while I work with anti/logs of semirings on a regular basis, I'm not entirely sure what the significance of anti/logs for matrices is. In part because they assume real/complex entries, whereas the types I tend to deal with have less structure than that. Also note, of course, that the anti/log only works for square matrices, and that not all matrices have a logarithm. So, if it does work out, it's terribly restricted compared to the other matrix operations.