The ancient Chinese had counting-board methods for solving systems of linear equations which correspond very closely to the modern determinant and Gaussian elimination methods. The earliest source for this (and in fact for Chinese mathematics altogether) is a set of bamboo strips found in a tomb dating from the year -186. The mathematics in question was probably already centuries old by then. Some of these strips are photographically reproduced on p. 29.
The strips explain how to solve two linear equations in two unknowns, for example:
"In dividing coins, if each person receives 2, there is a surplus of 3; if each person receives 3, there is a shortage of 2. It is asked how many persons and coins are there?" (p. 46)
"[Fine] rice costs 3 coins for 2 dou; coarse rice costs 2 coins for 3 dou. Now given 10 dou of coarse and fine rice [combined] that is sold for 13 coins, how much coarse rice and fine rice is there?" (p. 53)
The solutions correspond to the modern determinant method.
For systems of higher order the Chinese used the Gaussian elimination method. This method is described in the first-century treatise Nine Chapters on the Mathematical Arts. A sample problem:
"[We are to ascend a mountain carrying a weight of 40 dan] given one superior horse, two common horses, and three inferior horses. ... The superior horse together with one common horse, the [group of two] common horses together with one inferior horse, and the [group of three] inferior horses together with one superior horse, are all able to ascend. Problem: How much weight do the superior horse, common horse, and inferior horse each have the strength to pull?" (p. 165)
The solution is based on forming the "coefficient matrix" on the counting board and then bringing it into triangular form by row manipulations. One is reminded of those Mancala board games where marbles are placed in a grid of pits on a wooden board---a delightfully concrete version of a matrix, and one very well suited for these kinds of computations.