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:

$\displaystyle \left[\begin{array}{cc} a & b \\ [2pt] d & e \end{array}\right] \...
..._2 \end{array}\right] = \left[\begin{array}{c} c \\ [2pt] f \end{array}\right]
$

This can be written in higher level form as

$\displaystyle \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

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

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

$\displaystyle \left[\begin{array}{cc} a & b \\ [2pt] d & e \end{array}\right]^{...
...rac{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 [25]. An initial introduction to matrices and linear algebra can be found in [47].


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