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 |
|1545 Cardan: Cramer rule for 2x2 matrices. |
|1683 Seki and Leibnitz independently first appearance of Determinants |
|1750 Cramer (1704-1752) rule for solving systems of linear equations using determinants |
|1764 Bezout rule to determine determinants|
|1772 Laplace expansion of determinants|
|1801 Gauss first introduces determinants |
|1812 Cauchy multiplication formula of determinant. Independent of Binet|
|1812 Binet (1796-1856) discovered the rule det(AB) = det(A) det(B) |
|1826 Cauchy Uses term "tableau" for a matrix |
|1844 Grassman, geometry in n dimensions , (50 years ahead of its epoch [14 p. 204-205]|
|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  but still in 3 dimensions |
|1888 Giuseppe Peano (1858-1932) axioms of abstract vector space |
A quote from [14 p. 205]
The invention of matrices illustrates once more the power and suggestiveness of a well-devised 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
 Association J. Binet, mentions J. Shallit, Analysis of the Euclidean Algorithm, Historia Mathematica 21 (1994), 401-419.
 Grattan-Guinness, Convolutions in French Mathematics, 1800-1830 (Basel, 1990). WID-LC QA27.F8 G73 1990
 Vachov, D. Anniversaries in mathematics history for 1986. (Bulgarian) Translated in Proc. Steklov Inst. Math. 1990, no. 1, 279-284.
 Fiz.-Mat. Spis. B\cdprime lgar. Akad. Nauk. 29(62) (1987), no. 2, 118--120. Cabot Science Library PER 3740
Binet at MacTutor
Matrices and determinants at MacTutor
Abstract linear spaces at MacTutor
Wikipedia: Cramers rule
J. Tvrdá, On the origin of the theory of matrices, Acta Historiae Rerum Naturalium necnon Technicarum (Prague, 1971), 335-354. Widener Library Info Aus 80037.5
 E. Knobloch, Der Beginn der Determinantentheorie, Leibnizens nachgelassene Studien zum Determinantenkalkül (Hildesheim, 1980).
 A.E. Malykh, Development of the general theory of determinants up to the beginning of the nineteenth century (Russian), Mathematical analysis (Leningrad, 1990), 88-97.
 Nicolas Bourbaki: Elements of the history of mathematics, 1965 Scan.
 Bell: Toward mathematical structure, Scanned pages.
 Thomas Muir, Contributions to the History of determinants, (review) Nature, 126, 839, 1930 [Scan]
 [added Aug 1, 2014] Binet's 1812 paper (Thanks to Christoph Vignat) [PDF].