# Elementary matrix

(Redirected from Elementary row operation)

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group of invertible matrices. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. The acronym "ERO" is commonly used for "elementary row operations".[citation needed]

Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss-Jordan elimination to further reduce the matrix to reduced row echelon form.

## Elementary row operations

There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):

Row switching
A row within the matrix can be switched with another row.
${\displaystyle R_{i}\leftrightarrow R_{j}}$
Row multiplication
Each element in a row can be multiplied by a non-zero constant.
${\displaystyle kR_{i}\rightarrow R_{i},\ {\mbox{where }}k\neq 0}$
A row can be replaced by the sum of that row and a multiple of another row.
${\displaystyle R_{i}+kR_{j}\rightarrow R_{i},{\mbox{where }}i\neq j}$

If E is an elementary matrix, as described below, to apply the elementary row operation to a matrix A, one multiplies the elementary matrix on the left, E⋅A. The elementary matrix for any row operation is obtained by executing the operation on the identity matrix.

### Row-switching transformations

The first type of row operation on a matrix A switches all matrix elements on row i with their counterparts on row j. The corresponding elementary matrix is obtained by swapping row i and row j of the identity matrix.

${\displaystyle T_{i,j}={\begin{bmatrix}1&&&&&&&\\&\ddots &&&&&&\\&&0&&1&&\\&&&\ddots &&&&\\&&1&&0&&\\&&&&&&\ddots &\\&&&&&&&1\end{bmatrix}}\quad }$
So Tij⋅A is the matrix produced by exchanging row i and row j of A.

#### Properties

• The inverse of this matrix is itself: Tij−1=Tij.
• Since the determinant of the identity matrix is unity, det[Tij] = −1. It follows that for any square matrix A (of the correct size), we have det[TijA] = −det[A].

### Row-multiplying transformations

The next type of row operation on a matrix A multiplies all elements on row i by m where m is a non-zero scalar (usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the ith position, where it is m.

${\displaystyle D_{i}(m)={\begin{bmatrix}1&&&&&&\\&\ddots &&&&&\\&&1&&&&\\&&&m&&&\\&&&&1&&\\&&&&&\ddots &\\&&&&&&1\end{bmatrix}}\quad }$
So Di(m)⋅A is the matrix produced from A by multiplying row i by m.

#### Properties

• The inverse of this matrix is: Di(m)−1 = Di(1/m).
• The matrix and its inverse are diagonal matrices.
• det[Di(m)] = m. Therefore for a square matrix A (of the correct size), we have det[Di(m)A] = m det[A].

The final type of row operation on a matrix A adds row j multiplied by a scalar m to row i. The corresponding elementary matrix is the identity matrix but with an m in the (i,j) position.

${\displaystyle L_{i,j}(m)={\begin{bmatrix}1&&&&&&&\\&\ddots &&&&&&\\&&1&&&&&\\&&&\ddots &&&&\\&&m&&1&&\\&&&&&&\ddots &\\&&&&&&&1\end{bmatrix}}}$
So Li,j(m)⋅A is the matrix produced from A by adding m times row j to row i.

#### Properties

• These transformations are a kind of shear mapping, also known as a transvections.
• Lij(m)−1 = Lij(−m) (inverse matrix).
• The matrix and its inverse are triangular matrices.
• det[Lij(m)] = 1. Therefore, for a square matrix A (of the correct size) we have det[Lij(m)A] = det[A].
• Row-addition transforms satisfy the Steinberg relations.