Elementary matrix

Definition

Let $F$ be a field and let $n \ge 1$ be an integer. An $n \times n$ elementary matrix over $F$ is a matrix that results from the identity matrix $I_n$ by one elementary row operation. There are three types:

1. $P_{ij}$, for $i \ne j$: exchange rows $i$ and $j$ of $I_n$.
2. $D_i(c)$, for $c \ne 0$: multiply row $i$ of $I_n$ by $c$.
3. $T_{ij}(c)$, for $i \ne j$ and $c \in F$: add $c$ times row $j$ of $I_n$ to row $i$.

Remarks

For each $n \times p$ matrix $A$, the product $EA$ is the result of the same row operation on $A$. The product $AE$ performs the matching column operation on $A$. Each elementary matrix is invertible, and its inverse is an elementary matrix of the same type. Each invertible matrix over a field is a product of elementary matrices.

Explanations

L1 human

An elementary matrix stores one row operation as a matrix. To do the operation on $A$, we multiply $A$ on the left by this matrix. Then a whole elimination becomes a product of elementary matrices.