Elementary Operations on Matrices

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. interchanges of two rows of $M$ ,
2. multiply a row of $M$ by a non-zero element of $R$ ,
3. add a (constant) multiple of a row of $M$ to another row of $M$ .

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. $\begin{pmatrix} a & b \\ e & f \\ c & d \end{pmatrix}$ is obtained by interchanging rows 2 and 3 of $M$ ,
2. $\begin{pmatrix} a & b \\ rc & rd \\ e & f \end{pmatrix}$ is obtained by multiplying $r\ne 0$ to the second row of $M$ ,
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 $k$ .

Elementary Matrices

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. transposition matrix $T_{ij}$ : an matrix obtained from $I_n$ with rows $i$ and $j$ switched,
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. row replacement matrix $E_{ij}(s)$ : $I_n + s U_{ij}$ , where $s\in R$ and $U_{ij}$ is a matrix unit with $i\ne j$ .

For example, among the $3\times 3$ matrices, we have $$T_{12} = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad D_3(r) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & r \end{pmatrix},\quad\mbox{and}\quad E_{32}(s) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & s & 1 \end{pmatrix}$$

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 $$E_c^k(A)=\lbrace AE \mid E \in \mathbb{E}^k(n) \rbrace \qquad \mbox{and} \qquad E_r^k(A)=\lbrace EA \mid E \in \mathbb{E}^k(m) \rbrace.$$
• 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.

Remark. 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.