Solving Linear Equations Using Matrices

Consider the linear system of equations

$$\begin{eqnarray*} a x_1 + b x_2 &=& c \\ d x_1 + e x_2 &=& f \end{eqnarray*}$$

in matrix form:

$$\left[\begin{array}{cc} a & b \\ [2pt] d & e \end{array}\right] \left[\begin{array}{c} x_1 \\ [2pt] x_2 \end{array}\right] = \left[\begin{array}{c} c \\ [2pt] f \end{array}\right]$$

This can be written in higher level form as

$$\mathbf{A}\underline{x}= \underline{b},$$

where $\mathbf{A}$ denotes the two-by-two matrix above, and $\underline{x}$ and $\underline{b}$ denote the two-by-one vectors. The solution to this equation is then

$$\underline{x}= \mathbf{A}^{-1}\underline{b}= \left[\begin{array}{cc} a & b \\ [2pt] d & e \end{array}\right]^{-1}\left[\begin{array}{c} c \\ [2pt] f \end{array}\right].$$

The general two-by-two matrix inverse is given by

$$\left[\begin{array}{cc} a & b \\ [2pt] d & e \end{array}\right]^{-1} = \frac{1}{ae-bd}\left[\begin{array}{cc} e & -b \\ [2pt] -d & a \end{array}\right]$$

and the inverse exists whenever $ae-bd$ (which is called the determinant of the matrix $\mathbf{A}$ ) is nonzero. For larger matrices, numerical algorithms are used to invert matrices, such as used by Matlab based on LINPACK [26]. An initial introduction to matrices and linear algebra can be found in [49].