# A Classic from China: The Nine Chapters - Matrices

Author(s):
Randy K. Schwartz (Schoolcraft College)

The Chinese counting board, with its grid of square cells, was also useful for storing and manipulating rows and columns of numbers, rather than rows and columns of single digits. Such a numerical grid was called a fangcheng, literally “divided rectangle”. Much later in the West, this would be called a matrix.

The Chinese were many centuries in advance of the rest of the world in using matrices to solve systems of linear equations. The coefficients of each equation were stored in one column, and the columns were filled from right to left. The numbers were then manipulated using the same types of operations described above for problems of excess and deficit: multiplying or dividing a column by a given number, adding or subtracting two columns, etc. To solve a system of linear equations, the rectangle of coefficients was reduced to triangular and then diagonal form, in a process identical to what Europeans would later call Gaussian Elimination.

The first problem in Chapter 8 involves the harvest of three different grades of rice: 