In particular, determinants are born from matrices. Sylvester used grids of numbers to organize the calculation of determinants, which were useful in solving systems of n equations with n unknowns. Matrix multiplication and addition, or even matrix-vector equations, wouldn't come for another few decades.
Determinants are a rather old concept, named for their utility in determining whether or not a given system of equations has a unique solution. I believe determinants of general systems of 3 equations and 3 unknowns were known by Chinese mathematicians two millenia ago.
(My students seemed to be rather uninterested, or maybe more likely bemused, in the etymology.)
Calculating determinants was so important that even Charles Dodgson (a.k.a. Lewis Carrol) had a technique that he called condensation to evaluate them, a technique which had the benefit that the intermediate divisions would help you check your work in the event you expected the final determinant to be an integer.