- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.18 Computation techniques
$\quad$ The Gauss-Jordan elimination method consists of applying three basic types of operations on a linear system:
$\quad$ (1) Interchanging two equations;
$\quad$ (2) Multiplying all the terms of an equation by a nonzero scalar;
$\quad$ (3) Adding one equation to a multiple of another.
Each time we perform one of these operations on the system, we obtain a new system having exactly the same solutions. Two such systems are called equivalent.
$\quad$ Example. $\quad$ A system with a unique solution. Consider the system \begin{align*} 2x - 5y + 4z &= -3 \\ x - 2y + z &= 5 \\ x - 4y + 6z &= 10. \end{align*} This particular system has a unique solution $x = 124, y=75, z=31,$ which can be obtained by the Gauss-Jordan elimination process. To avoid writing variable names repeatedly, we can work with an augmented matrix: \begin{align*} \begin{bmatrix} \begin{array}{ccc|c} 2 & -5 & 4 & -3 \\ 1 & -2 & 1 & 5 \\ 1 & -4 & 6 & 10 \end{array} \end{bmatrix} \end{align*} obtained by adjoining the right-hand members of the system to the coefficient matrix. The three operations mentioned in the Gauss-Jordan elimination process are called row operations. At any point we can put the letters $x, y, z$ back into the equation and insert the equals sign along the vertical line to obtain equations. Our goal is to arrive at the augmented matrix \begin{bmatrix} \begin{array}{ccc|c} 1 & 0 & 0 & 124 \\ 0 & 1 & 0 & 75 \\ 0 & 0 & 1 & 31 \end{array} \end{bmatrix}