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Solving Linear Equations Using Matrices

Consider the linear system of equations

a x_1 + b x_2 &=& c \\
d x_1 + e x_2 &=& f

in matrix form:

$\displaystyle \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

$\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}{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

$\displaystyle \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].

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-02-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University