IfA is an invertiblen×n matrix with entries in the finite field F_q, letT _n (A) be its minimum period or exponent, i.e. its order as an element of the general linear group GL(n,q). The main result is, roughly, that $T_n (A) = q^{n - } (log n)^{2 + 0(1)} $ for almost everyA.