Gauss-Jordan elimination

(noun)

In linear algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. It is a variation of Gaussian elimination.

Examples of Gauss-Jordan elimination in the following topics:

  • Further Simplification of Matrices

    • Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations.
    • In linear algebra, Gauss–Jordan elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations .
    • Gauss–Jordan elimination goes a step further by placing zeros above and below each pivot; such matrices are said to be in reduced row echelon form.
    • Gauss-Jordan elimination, like Gaussian elimination, is used for inverting matrices and solving systems of linear equations.
    • A matrix is in reduced row echelon form (also called row canonical form) if it is the result of a Gauss–Jordan elimination.
  • Simplifying Matrices With Row Operations

    • Using elementary operations, Gaussian elimination reduces matrices to row echelon form.
    • By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to a row echelon form.  
    • Using elementary row operations at the end of the first part (Gaussian elimination, zeros only under the leading 1) of the algorithm:
    • At the end of the algorithm, if the Gauss–Jordan elimination (zeros under and above the leading 1) is applied:
  • Matrices and Row Operations

    • Also, when solving a system of linear equations by the elimination method, row multiplication would be the same as multiplying the whole equation by a number to obtain additive inverses so that a variable cancels.  
    • Finally, row addition is also the same as the elimination method, when one chooses to add or subtract the like terms of the equations to obtain the variable.  
    • There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are Gaussian elimination and Gauss-Jordan elimination.  
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