To give the history of linear algebra is a task that is as important as it is difficult  Nicolas Bourbaki, 1984 
200 BC: Han dynasty, coefficients are written on a counting board [6] 
1545 Cardan: Cramer rule for 2x2 matrices. [6] 
1683 Seki and Leibnitz independently first appearance of Determinants [6] 
1750 Cramer (17041752) rule for solving systems of linear equations using determinants [8] 
1764 Bezout rule to determine determinants 
1772 Laplace expansion of determinants 
1801 Gauss first introduces determinants [6] 
1812 Cauchy multiplication formula of determinant. Independent of Binet 
1812 Binet (17961856) discovered the rule det(AB) = det(A) det(B) [1] 
1826 Cauchy Uses term "tableau" for a matrix [6] 
1844 Grassman, geometry in n dimensions [14], (50 years ahead of its epoch [14 p. 204205] 
1850 Sylvester first use of term "matrix" (matrice=pregnant animal in old french or matrix=womb in latin as it generates determinants) 
1858 Cayley matrix algebra [7] but still in 3 dimensions [14] 
1888 Giuseppe Peano (18581932) axioms of abstract vector space [12] 
A quote from [14 p. 205]
The invention of matrices illustrates once more the power and suggestiveness of a welldevised notation;
it also exemplifies the fact, which some mathematicians are reluctant to admit, that a trivial
notational device may be the germ of a vast theory having innumerable applications.
Cayley told Tait in 1894 what led him to matrices: "I certainly did not get the notion of a matrix in any way through
quaternions: it was either directly from that of a determinant; or as a convenient mode of expression of the equations

Symbolizing this linear transformation on two independent variables by the square array

[1] Association J. Binet, mentions J. Shallit, Analysis of the Euclidean Algorithm, Historia Mathematica 21 (1994), 401419.
[2] GrattanGuinness, Convolutions in French Mathematics, 18001830 (Basel, 1990). WIDLC QA27.F8 G73 1990
[3] Vachov, D. Anniversaries in mathematics history for 1986. (Bulgarian) Translated in Proc. Steklov Inst. Math. 1990, no. 1, 279284.
[4] Fiz.Mat. Spis. B\cdprime lgar. Akad. Nauk. 29(62) (1987), no. 2, 118120. Cabot Science Library PER 3740
[6]Matrices and determinants at MacTutor
[7]Abstract linear spaces at MacTutor
[9]J. Tvrdá, On the origin of the theory of matrices, Acta Historiae Rerum Naturalium necnon Technicarum (Prague, 1971), 335354. Widener Library Info Aus 80037.5
[10] E. Knobloch, Der Beginn der Determinantentheorie, Leibnizens nachgelassene Studien zum Determinantenkalkül (Hildesheim, 1980).
[11] A.E. Malykh, Development of the general theory of determinants up to the beginning of the nineteenth century (Russian), Mathematical analysis (Leningrad, 1990), 8897.
[12]Peano
[13] Nicolas Bourbaki: Elements of the history of mathematics, 1965 Scan.
[14] Bell: Toward mathematical structure, Scanned pages.
[15] Thomas Muir, Contributions to the History of determinants, (review) Nature, 126, 839, 1930 [Scan]
[16] [added Aug 1, 2014] Binet's 1812 paper (Thanks to Christoph Vignat) [PDF].