Let K be a proper (i.e., closed, pointed, full convex) cone in R n . An n × n matrix A is said to be K -primitive if there exists a positive integer k such that A k ( K ⧹ { 0 } ) ⊆ int K ; the least such k is referred to as the exponent of A and is denoted by γ ( A ) . For a polyhedral cone K , the maximum value of γ ( A ) , taken over all K -primitive matrices A , is called the exponent of K and is denoted by γ ( K ) . It is proved that the maximum value of γ ( K ) as K runs through all n -dimensional minimal cones (i.e., cones having n + 1 extreme rays) is n 2 - n + 1 if n is odd, and is n 2 - n if n is even, the maximum value of the exponent being attained by a minimal cone with a balanced relation for its extreme vectors. The K -primitive matrices A such that γ ( A ) attain the maximum value are identified up to cone-equivalence modulo positive scalar multiplication.