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Inlinear algebra, twomatrices arerow equivalent if one can be changed to the other by a sequence ofelementary row operations. Alternatively, twom × n matrices are row equivalent if and only if they have the samerow space. The concept is most commonly applied to matrices that representsystems of linear equations, in which case two matrices of the same size are row equivalent if and only if the correspondinghomogeneous systems have the same set of solutions, or equivalently the matrices have the samenull space.

Because elementary row operations are reversible, row equivalence is anequivalence relation. It is commonly denoted by atilde (~).^[1]

There is a similar notion ofcolumn equivalence, defined by elementary column operations; two matrices are column equivalent if and only if their transpose matrices are row equivalent. Two rectangular matrices that can be converted into one another allowing both elementary row and column operations are called simplyequivalent.

Anelementary row operation is any one of the following moves:

1. Swap: Swap two rows of a matrix.
2. Scale: Multiply a row of a matrix by a nonzero constant.
3. Pivot: Add a multiple of one row of a matrix to another row.

Two matricesA andB arerow equivalent if it is possible to transformA intoB by a sequence of elementary row operations.

The row space of a matrix is the set of all possiblelinear combinations of its row vectors. If the rows of the matrix represent asystem of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. Twom × n matrices are row equivalent if and only if they have the same row space.

For example, the matrices

are row equivalent, the row space being all vectors of the form. The corresponding systems of homogeneous equations convey the same information:

${\displaystyle \begin{array}{c}x=0\\ y+z=0\end{array}\text{and}\begin{array}{c}x=0\\ x+y+z=0.\end{array}}$

In particular, both of these systems imply every equation of the form${\displaystyle ax+by+bz=0.}$

The fact that two matrices are row equivalent if and only if they have the same row space is an important theorem in linear algebra. The proof is based on the following observations:

1. Elementary row operations do not affect the row space of a matrix. In particular, any two row equivalent matrices have the same row space.
2. Any matrix can bereduced by elementary row operations to a matrix inreduced row echelon form.
3. Two matrices in reduced row echelon form have the same row space if and only if they are equal.

This line of reasoning also proves that every matrix is row equivalent to a unique matrix with reduced row echelon form.

• Because thenull space of a matrix is theorthogonal complement of therow space, two matrices are row equivalent if and only if they have the same null space.
• Therank of a matrix is equal to thedimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number ofpivots in the reduced row echelon form.
• A matrix isinvertible if and only if it is row equivalent to theidentity matrix.
• MatricesA andB are row equivalent if and only if there exists an invertible matrixP such thatA=PB.^[2]

• Elementary row operations
• Row space
• Basis (linear algebra)
• Row reduction
• (Reduced) row echelon form

• Axler, Sheldon Jay (1997),Linear Algebra Done Right (2nd ed.), Springer-Verlag,ISBN 0-387-98259-0
• Lay, David C. (August 22, 2005),Linear Algebra and Its Applications (3rd ed.), Addison Wesley,ISBN 978-0-321-28713-7
• Meyer, Carl D. (February 15, 2001),Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM),ISBN 978-0-89871-454-8, archived fromthe original on March 1, 2001
• Poole, David (2006),Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole,ISBN 0-534-99845-3
• Anton, Howard (2005),Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
• Leon, Steven J. (2006),Linear Algebra With Applications (7th ed.), Pearson Prentice Hall
• Roman, Steven (2008).Advanced Linear Algebra.Graduate Texts in Mathematics. Vol. 135 (3rd ed.). Springer Science+Business Media, LLC.ISBN 978-0-387-72828-5.

The WikibookLinear Algebra has a page on the topic of:Row Equivalence

• Linear algebra
• Matrices (mathematics)
• Equivalence (mathematics)

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