Dedekind raised the question of computing the determinant of the multiplication table of a finite group (regarding the group elements as commuting indeterminates). The abelian case was well-understood. Frobenius was a master of determinants and created group representation theory in order to answer Dedekind's question. Frobenius' approach to group representations was based on determinants. This is explained by T. Hawkins in Arch. History Exact Sci. 27 (1970/71), 142-170; 8 (1971/72), 243-287; 12 (1974), 217-243.
I should also point out that the evaluation of determinants is alive and well within combinatorics. Often a number or generating function can be expressed as a determinant. This is considered a "nice" answer because determinants can sometimes be evaluated explicitly, and in any event can be evaluated quickly and have many other useful properties. See for instance the work of Krattenthaler, especially http://www.mat.univie.ac.at/~kratt/artikel/detsurv.html and the sequel http://www.mat.univie.ac.at/~kratt/artikel/detcomp.html.