The earliest known mention of matrices to solve systems of linear equations occurred in Jiuzhang suanshu or Nine Chapters on the Mathematical Art, a Chinese mathematical treatise from 100 to 200 BC. The treatise not only describe the process of Naive Gaussian Elimination to solve linear equations but also introduces the concept of a determinant. The work most likely describes methods from older mathematical treatises lost when emperor Shih Huang Ti organized a great book burning in 238BC.

Unfortunately, matrices were not widely used until the 16th century and determinants did not reappear until the late 1600s when they were rediscovered independently by Seki Kowa in Japan and Gottfried Leibniz in Germany. Gaussian elimination would not be described by Carl Friedrich Gauss until 1803 when he used the technique to study to orbit of the asteroid Pallas.

The term ‘matrix’ was coined by the British mathematician John Joseph Sylvester in 1850. Incidentally, Sylvester’s career at the University of Virginia was cut short when he came to believe that he killed a student who was reading a newspaper during one of his lectures after hitting him with a sword stick. The student was merely in cardiac shock but Sylvester fled the country before he found out. A sword stick or cane sword was a cane with a concealed blade which was popular among the European upper classes in the 18^th and early 19^th centuries.

Sources:

http://ualr.edu/lasmoller/matrices.html

http://www-groups.dcs.st-and.ac.uk/%7Ehistory/HistTopics/Matrices_and_determinants.html

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Nine_chapters.html