Let X be a matrix drawn uniformly at random from GL_n(F_q), then one may conjecture that traces of powers of such matrices Tr(X^k) should have an asymptotically uniform distribution F_q. Further, one may wonder how robust this phenomenon is and how large can k be. On the one hand, this question is an analogue of a classic problem from random matrix theory, and, on the other hand, it is intimately related to short character sums over function fields with the power k serving as the conductor. In our work, we prove that the distribution of Tr(X^k) is indeed asymptotically uniform and that the respective short interval character sums exhibit cancellation for k = q^{o(n^2)}. This is a much wider range than one could hope to obtain for general characters, and in fact, this phenomenon seems to have no analogue over the integers. This is joint work with Ofir Gorodetsky.