RowReduce

RowReduce[m]

gives the row‐reduced form of the matrix m.

Details and Options

  • RowReduce performs a version of Gaussian elimination, adding multiples of rows together so as to produce zero elements when possible. The final matrix is in reduced row echelon form.
  • If m is a non‐degenerate square matrix, RowReduce[m] is IdentityMatrix[Length[m]]. »
  • If m is a sufficiently non‐degenerate rectangular matrix with rows and more than columns, then the first columns of RowReduce[m] will form an identity matrix. »
  • RowReduce works on both numerical and symbolic matrices.
  • The following options can be given:
  • MethodAutomaticmethod to use
    Modulus0integer modulus to use
    ToleranceAutomaticnumerical tolerance to use
    ZeroTestAutomaticfunction to test whether matrix elements should be considered to be zero
  • RowReduce[m,Modulus->n] performs row reduction modulo n. »
  • RowReduce[m,ZeroTest->test] evaluates test[m[[i,j]]] to determine whether matrix elements are zero.
  • Possible settings for the Method option include "CofactorExpansion", "DivisionFreeRowReduction", and "OneStepRowReduction". The default setting of Automatic switches among these methods depending on the matrix given.

Examples

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Basic Examples  (3)

Do row reduction on a square matrix:

Do row reduction on a rectangular matrix:

Scope  (2)

m is an exact 4×4 matrix:

Use exact arithmetic for the Gaussian elimination:

With machine‐number arithmetic, the leading zeros and ones are still exact:

With higher‐precision arithmetic, the leading zeros and ones are still exact:

Row reduction for a random complex 3×4 matrix:

Options  (2)

Modulus  (1)

m is a 3×3 matrix of integers between 0 and 4:

Row reduction of m in ordinary arithmetic:

Row reduction of m in arithmetic modulo 5:

Tolerance  (1)

m is an ill-conditioned matrix:

In exact arithmetic, m is clearly non-degenerate:

With machine arithmetic, the default is to consider elements that are too small as zero:

With zero tolerance, even small terms may be taken into account:

With an augmented matrix, you can see how possible solution components are amplified:

Applications  (2)

m is a matrix and b is a vector:

Form an augmented matrix :

Do row reduction on the augmented matrix:

The last column is the solution of :

Do it for another right‐hand side:

There is no solution to since there is a leading 1 in the last column:

m is a symbolic matrix:

Find the symbolic inverse:

This is the equivalent to what is given by Inverse:

Properties & Relations  (3)

m is a non-degenerate square matrix:

RowReduce[m] is IdentityMatrix[Length[m]]:

Augment m with the 3×3 identity matrix:

Do row reduction on the augmented matrix:

The last three columns of the result are Inverse[m]:

m is a 3×5 matrix:

m has its maximal MatrixRank of 3:

The first 3 columns of RowReduce[m] are the 3×3 identity matrix:

m is a degenerate square matrix:

Augment m with the 4×4 identity matrix:

Do row reduction on the augmented matrix:

The last 4 columns of any row with the leading one beyond the 4^(th) column is in the null space:

Even though the vectors are not the same, they are a basis for the same space:

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)