3.4. Using Matrix Algebra to Solve Linear Equations

We now turn our attention to an important application of matrix algebra, which is to solve sets of linear equations.

3.4.1. Representation

Systems of linear equations are conveniently represented by matrices. Consider the set of linear equations:

3x + 2y + z = 5

–8x + y + 4z = –2

9x + 0.5y + 4z = 0.9

We can represent this set of equations by the matrix

where the position of a number in the matrix implicitly identifies it as either a coefficient of a variable or a value on the right-hand side. This representation can be used for any set of linear equations. If the rightmost column is 0, the system is said to be homogeneous. The submatrix corresponding to the left-hand side of the linear equations is called the coefficient matrix.