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SciTech-Math-AdvancedAlgebra- Cramer' Rule (Gabriel Cramer (1704–1752)) + Gauss-Jordan Method

发布时间 2024-01-09 20:48:29作者: abaelhe

2.2: Systems of Linear Equations and the Gauss-Jordan Method

Learning Objectives
In this section you will learn to

  • Represent a system of linear equations as an augmented matrix
  • Solve the system using elementary row operations.

In this section, we learn to solve systems of linear equations using a process called the Gauss-Jordan method.
The process:

  1. begins by first expressing the system as a matrix,
    we expressed the system of equations as ??=? , where:

    • ? represented the coefficient matrix,
    • ? the matrix of constant terms.
    • [??] As an augmented matrix, we write the matrix as [??] .
    • It is clear that all of the information is maintained in this matrix form,
      and only the letters ? , ? and ? are missing. A student may choose to write ? , ? and ? on top of the first three columns to help ease the transition.

  2. and then reducing it to an equivalent system by simple row operations.

  3. The process is continued until the solution is obvious from the matrix.

The matrix that represents the system is called the augmented matrix,
and the arithmetic manipulation that is used to move from a system to a reduced equivalent system is called a row operation.

href="https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/09%3A_Systems_of_Equations_and_Inequalities/9.08%3A_Solving_Systems_with_Cramer's_Rule">https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_1e_(OpenStax)/09%3A_Systems_of_Equations_and_Inequalities/9.08%3A_Solving_Systems_with_Cramer's_Rule