# Determinant computation is equivalent to matrix powering

It has been claimed in this paper (page 2 last paragraph) that Matrix powering is equivalent to determinant computation.

https://www.cse.iitk.ac.in/users/manindra/algebra/depth-four.pdf

Does anybody why is this the case?

From another angle, Coppersmith–Winograd algorithm for matrix multiplication has complexity $$\mathcal O(n^{2.373})$$ and the same complexity is for determinant computation by fast multiplication.
Look at the paper by Stephen Cook (numbered $$[3]$$ in the references of the paper you have mentioned). There, in proposition $$5.2$$ in page $$13$$, he shows the "computational equivalence" between matrix powering and determinant computation (and other problems).