Sylvester coined the name "matrix", using the Latin for "womb" because he saw them as generators. (I didn't know either until this thread.)

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.

There are also printing matrices... I wonder which usage was developed first?

Looking at the OED, usage as mould or shape for something, including printing is much earlier. In 1526 Tyndale apparently used it in his gospel translation - "Every man chylde that first openeth the matrix shalbe called holy to the lorde"†. It's a little misleading to say Sylvester 'coined' the term, he just used an existing term for his particular purpose.

† this also suggests William Tyndale possibly wrote the earliest Transformers fanfic.