I am not sure I understand the question. Any matrix $A$ (integer or not, positive or not) has a Jordan canonical form $A = MJM^{-1},$ whereupon $A^n = M J^n J^{-1}.$$A^n = M J^n M^{-1}.$ If $A$ is integer and nonsingular, the biggest eigenvalue is at least $1$ in modulus (since the determinant is at least $1$ in absolute value). If it IS equal to $1$ in modulus, the sum of the elements will be polynomial, if it is greater than one, it will be exponential -- if there is a single eigenvalue of maximal modulus, it will be really exponential, otherwise at least there will be a positive density subsequence of $n$ for which it is. Which numbers can occur as eigenvalues of nonnegative integer matrices was answered by Doug Lind in:
Lind, D. A.(1-WA) The entropies of topological Markov shifts and a related class of algebraic integers. Ergodic Theory Dynam. Systems 4 (1984), no. 2, 283–300. 58F11 (15A48 28D20)
MR1149738 (92m:11117) Reviewed Lind, Douglas(1-WA) Matrices of Perron numbers. J. Number Theory 40 (1992), no. 2, 211–217. 11R06 (15A48 58F03)
