Let $\mathbb{M}$ be the set of all $m\times n$ matrices (over some commutative ring $R$). An operation on $\mathbb{M}$ is called an elementary row operation if it takes a matrix $M\in\mathbb{M}$, 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 $\mathbb{M}$ is an elementary operation if it is either an elementary row operation or elementary column operation.

For example, if $M=\begin{pmatrix}a&b\\ c&d\\ e&f\end{pmatrix}$, then the following operations correspond respectively to the three types of elementary row operations described above

1. 1.

   $\begin{pmatrix}a&b\\ e&f\\ c&d\end{pmatrix}$ is obtained by interchanging rows 2 and 3 of $M$,

2. 2.

   $\begin{pmatrix}a&b\\ rc&rd\\ e&f\end{pmatrix}$ is obtained by multiplying $r\neq 0$ to the second row of $M$,

3. 3.

   $\begin{pmatrix}a&b\\ c&d\\ sa+e&sb+f\end{pmatrix}$ 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 invertible. The inverse 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 $E_{c}^{k}(A)$ be the set of all matrices obtained from $A$ via an elementary column operation of type $k$, and $E_{r}^{k}(A)$ the set of all matrices obtained from $A$ via an elementary row operation of type $k$.

Now, assume $R$ has $1$. An $n\times n$ elementary matrix is a (square) matrix obtained from the identity matrix $I_{n}$ 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 $T_{{ij}}$: an matrix obtained from $I_{n}$ with rows $i$ and $j$ switched,

2. 2.

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

3. 3.

   row replacement matrix $E_{{ij}}(s)$: $I_{n}+sU_{{ij}}$, where $s\in R$ and $U_{{ij}}$ is a matrix unit with $i\neq j$.

For example, among the $3\times 3$ matrices, we have

For each positive integer $n$, let $\mathbb{E}^{k}(n)$ be the collection of all $n\times n$ elementary matrices of type $k$, where $k=1,2,3$.

Below are some basic properties of elementary matrices:

• $T_{{ij}}=T_{{ji}}$, and $T_{{ij}}^{2}=I_{n}$.

• $D_{i}(r)D_{i}(r^{{-1}})=I_{n}$, provided that $r^{{-1}}$ exists.

• $E_{{ij}}(s)E_{{ij}}(-s)=I_{n}$.

• $\det(T_{{ij}})=-1$, $\det(D_{i}(r))=r$, and $\det(E_{{ij}}(s))=1$.

• If $A$ is an $m\times n$ matrix, then

  $$E_{c}^{k}(A)=\{AE\mid E\in\mathbb{E}^{k}(n)\}\qquad\mbox{and}\qquad E_{r}^{k}(A)=\{EA\mid E\in\mathbb{E}^{k}(m)\}.$$

• Every non-singular matrix can be written as a product 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.

Remarks.

• 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.