Every student of linear algebra learns about the trace of a linear map. Its easiest (albeit not most conceptual) definition is: write the map as matrix, then the trace is the sum of the diagonal entries.
The question about origin and motivation of this terminology has been asked several times: https://math.stackexchange.com/q/1291981/96384, https://math.stackexchange.com/q/2593048/96384, https://math.stackexchange.com/q/3684703/96384.
However, none of them have a fully satisfying answer. What they seem to agree on is that the English "trace" is a translation of the German terminology Spur (which is cognate to English spoor, but indeed translates to trace or track): "Earliest Known Uses of Some of the Words of Mathematics" quotes H. L. Brose's 1922 translation of a book of Weyl's, which might be the first use of the English term "trace". But then the question remains,
In what (presumably German-language) source was the word "Spur" used first for this concept?
Attempt: At the bottom of page 6 of his (1901) doctoral thesis Ueber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, Issai Schur writes somewhat casually [emphasis added]:
dass zwei invariante Formen $T(A)$ und $T_1(A)$ dann und nur dann äquivalent sind, wenn ihre Spuren, d.h. die Summen der Glieder ihrer Hauptdiagonalen einander gleich sind.
Since Schur's doctoral advisor was Frobenius, I thought it natural to look for earlier occurrences in Frobenius' works. However, I learned that in Frobenius's three Sitzungsberichte papers from 1896 where he founded character theory, he actually does it via a different route, and I found no trace of a "Spur" in those. (For some overview on the history of representation and character theory, cf. T. Hawkins and K. Conrad.) Then, in Frobenius' 1897 paper Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, where he finally uses matrices, he does note that his characters turn out to be sums of diagonal elements (equation 5 in §4), and he emphasizes the importance of this especially in §6 (pages 15 and 16), but as far as I can tell, nowhere does he use the word "Spur"; instead, he writes out "die Summe der Diagonalelemente" twice.
So now I am almost tempted to believe that Schur might have casually introduced that terminology there? Of course I might have overlooked something in Frobenius' work of that time. Another guess would be Dedekind, who according to the mentioned historical sources played an important "inspirational" role in that time. I have not looked through any of his works yet. Surely from today's view, he would have had reason to be interested in "Spuren" already in his number theoretic works.
PS: In one of the linked questions above, a theory is put forward and credited to P.J. Cohen that supposedly, German "Spur" was a mistranslation of an English word "spur" supposedly used by Cayley (which would have been earlier than my guesses). This theory sounds rather unlikely to me, and according to a comment by Brian M. Scott to that question, "a search of the University of Michigan Historical Mathematics Collection turns up only two instance of spur in Cayley’s collected works, and they have to do with physical gears, not matrices."