elementary matrix

Elementary Operations on Matrices

Let 𝕄 be the set of all m×n matrices (over some commutative ring R). An operationMathworldPlanetmath on 𝕄 is called an elementary row operation if it takes a matrix M𝕄, and does one of the following:

  1. 1.

    interchanges of two rows of M,

  2. 2.

    multiply a row of M by a non-zero element of R,

  3. 3.

    add a (constant) multiple of a row of M to another row of M.

An elementary column operation is defined similarly. An operation on 𝕄 is an elementary operation if it is either an elementary row operation or elementary column operation.

For example, if M=(abcdef), then the following operations correspond respectively to the three types of elementary row operations described above

  1. 1.

    (abefcd) is obtained by interchanging rows 2 and 3 of M,

  2. 2.

    (abrcrdef) is obtained by multiplying r0 to the second row of M,

  3. 3.

    (abcdsa+esb+f) is obtained by adding to row 1 multiplied by s to row 3 of M.

Some immediate observation: elementary operations of type 1 and 3 are always invertiblePlanetmathPlanetmathPlanetmath. The inversePlanetmathPlanetmathPlanetmathPlanetmath of type 1 elementary operation is itself, as interchanging of rows twice gets you back the original matrix. The inverse of type 3 elementary operation is to add the negative of the multiple of the first row to the second row, thus returning the second row back to its original form. Type 2 is invertible provided that the multiplier has an inverse.

Some notation: for each type k (where k=1,2,3) of elementary operations, let Eck(A) be the set of all matrices obtained from A via an elementary column operation of type k, and Erk(A) the set of all matrices obtained from A via an elementary row operation of type k.

Elementary Matrices

Now, assume R has 1. An n×n elementary matrix is a (square) matrix obtained from the identity matrixMathworldPlanetmath In by performing an elementary operation. As a result, we have three types of elementary matrices, each corresponding to a type of elementary operations:

  1. 1.

    transposition matrix Tij: an matrix obtained from In with rows i and j switched,

  2. 2.

    basic diagonal matrix Di(r): a diagonal matrixMathworldPlanetmath whose entries are 1 except in cell (i,i), whose entry is a non-zero element r of R

  3. 3.

    row replacement matrix Eij(s): In+sUij, where sR and Uij is a matrix unit with ij.

For example, among the 3×3 matrices, we have


For each positive integer n, let 𝔼k(n) be the collectionMathworldPlanetmath of all n×n elementary matrices of type k, where k=1,2,3.

Below are some basic properties of elementary matrices:

  • Tij=Tji, and Tij2=In.

  • Di(r)Di(r-1)=In, provided that r-1 exists.

  • Eij(s)Eij(-s)=In.

  • det(Tij)=-1, det(Di(r))=r, and det(Eij(s))=1.

  • If A is an m×n matrix, then

    Eck(A)={AEE𝔼k(n)}  and  Erk(A)={EAE𝔼k(m)}.
  • Every non-singular matrix can be written as a productPlanetmathPlanetmath of elementary matrices. This is the same as saying: given a non-singular matrix, one can perform a finite number of elementary row (column) operations on it to obtain the identity matrix.


  • One can also define elementary matrix operations on matrices over general rings. However, care must be taken to consider left scalar multiplications and right scalar multiplications as separate operations.

  • The discussion above pertains to elementary linear algebra. In algebraic K-theory, an elementary matrix is defined only as a row replacement matrix (type 3) above.

Title elementary matrix
Canonical name ElementaryMatrix
Date of creation 2013-03-22 18:30:38
Last modified on 2013-03-22 18:30:38
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 15-01
Related topic MatrixUnit
Related topic GaussianElimination
Defines elementary operation
Defines elementary column operation
Defines elementary row operation
Defines basic diagonal matrix
Defines transposition matrix
Defines row replacement matrix