  We analyze the precision of the characteristic polynomial of an $n\times n$
p-adic matrix A using differential precision methods developed previously. When
A is integral with precision O(p^N), we give a criterion (checkable in time
O~(n^omega)) for $\chi$(A) to have precision exactly O(p^N). We also give a
O~(n^3) algorithm for determining the optimal precision when the criterion is
not satisfied, and give examples when the precision is larger than O(p^N).
