Counting sort initialization of the output array, each take time. Therefore, the time for the whole algorithm is the sum of the times for these steps, . Because it uses arrays of length and , the total space usage of the algorithm is also . For problem instances in which the maximum key value is significantly smaller than the number of items, counting sort can be highly space-efficient, as the only storage it uses other than its input and output arrays is the Count array which uses space . If each item to be sorted is itself an integer, and used as
Integer matrix not become excessively large (see condition number). Theorems from matrix theory that infer properties from determinants thus avoid the traps induced by ill conditioned ("nearly" zero determinant) real or floating point valued matrices. The inverse of an integer matrix formula_3 is again an integer matrix if and only if the determinant of formula_3 is exactly formula_5 or formula_6. Integer matrices of determinant formula_5 form the group formula_8, which has far-reaching applications in arithmetic and geometry. For formula_9, it is closely related to the modular group. The intersection of the integer matrices with the orthogonal group is the group of signed