Gauss-Jordan 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.