Who first invented/introduced the concept of the trace of a Matrix and Why? Could anyone give any information  about the invention of the concept of the trace of a Matrix, as this concept  is so important and useful in linear algebra. I searched on the internet, but found nothing on its origin.
 A: From Where did the German term "Spur" of a matrix come from? we learn of "die Spur". You can also review Motivation for the term "trace" in linear algebra.
Using that tidbit and this website, we find:
Trace (of a matrix) is a translation of the German die Spur (related to the English word "spoor.") 
In his 1922 translation of H. Weyl’s Raum, Zeit, Materie (Space-Time-Matter) H. L. Brose writes: "the trace (spur) of a matrix." 
Some writers in English preferred the term "spur," e.g. A. C. Aitken Determinants and Matrices (9th edition 1956) writes "the spur or trace of A. We shall denote it by sp A."
A: To upgrade Moo's answer with what, for all we know, is the German-language prehistory of "trace":
Dedekind introduced what in modern terminology is the field trace as "Spur" on page 5 of his article Über die Discriminanten endlicher Körper. (In: Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, 1882), as a third rational invariant for algebraic numbers, after the already known discriminants and norms in algebraic field extensions.
Dedekind did notice that this Spur can be computed via summing the diagonal elements of a matrix, but did not make much use of that; he mostly viewed it as a sum of Galois conjugates, and discussed it at length in his Supplement XI to the fourth edition (1894) of Dirichlet's Vorlesungen über Zahlentheorie. By 1897, this use had been adopted by an algebra textbook of Weber's and in an article of Hensel's.
When Frobenius founded character theory in 1896, he did so without using matrices at all; when he brought matrices into play in 1897, he noticed that his characters come out as sums of diagonal elements of matrices, but did not give that a special name; only in his 1899 report Über die Composition der Charaktere einer Gruppe he writes (bottom of first page):

Nennt man nach dem Vorgange von DEDEKIND die Summe der Diagonalelemente einer Substitution oder Matrix ihre Spur, [...]

that is, he ("following Dedekind") takes the sum-of-diagonal definition for any matrix, not just ones coming from field elements, and might have been the first to do so. Frobenius' doctoral student Schur followed suit on page 6 of his (1901) thesis Ueber eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen, and certainly it spread widely after this.
A: Cayley (in particular) , and other British Mathematicians, wrote early papers on matrices, 
he even used [a b, c d] as notation for a typical 2x2 matrix, which we still use today.
So probably trace is in these early papers. Others involved were Sylvester, Kirkman. 
