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1.3.3 Elimination Method (Gaussian Elimination)

Solve systems of linear equations by adding, subtracting, or multiplying equations to eliminate variables.

定义

The Elimination Method, also known as Gaussian Elimination, is a systematic algebraic technique for solving systems of linear equations. The method involves performing elementary row operations on the equations to eliminate variables progressively, reducing the system to a simpler form (typically row echelon form or reduced row echelon form) from which solutions can be easily obtained. The three elementary row operations are: (1) multiplying an equation by a non-zero constant, (2) adding or subtracting one equation from another, and (3) swapping the positions of two equations. These operations preserve the solution set of the system while simplifying the structure. The goal is to create a triangular or diagonal pattern of coefficients that allows for back-substitution to find the values of all variables. For a system of \(m\) equations with \(n\) unknowns, the method can determine whether the system has a unique solution, infinitely many solutions, or no solution.

核心公式

  • \(\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} & | & b_1 \ a_{21} & a_{22} & \cdots & a_{2n} & | & b_2 \ \vdots & \vdots & \ddots & \vdots & | & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} & | & b_m \end{bmatrix}\)
  • \(R_i \leftarrow cR_i\) (multiply row \(i\) by non-zero constant \(c\))
  • \(R_i \leftarrow R_i + kR_j\) (add \(k\) times row \(j\) to row \(i\))
  • \(R_i \leftrightarrow R_j\) (swap rows \(i\) and \(j\))
  • \(\begin{bmatrix} 1 & 0 & 0 & | & x \\ 0 & 1 & 0 & | & y \\ 0 & 0 & 1 & | & z \end{bmatrix}\) (reduced row echelon form for unique solution)

易错点

  • ⚠️ Forgetting to apply the same operation to the constant term (right side) of the equation when performing row operations, leading to incorrect solutions
  • ⚠️ Making arithmetic errors when multiplying equations by constants or adding/subtracting equations, especially with negative numbers and fractions
  • ⚠️ Stopping the elimination process prematurely before achieving row echelon form, or failing to recognize when a system has infinitely many solutions or no solution (inconsistent systems)
  • ⚠️ Incorrectly performing back-substitution by not carefully tracking which variables have been eliminated and substituting in the wrong order