Every elementary matrix of the first kind is a commutator of invertible matrices

Statement

Suppose $k$ is a field and $n$ is a natural number. For $\lambda \in k$ and $i\neq j$ elements of $\{1,2,3,\dots ,n\}$, denote by $E_{ij}(\lambda )$ the matrix with $1$s on the diagonal, $\lambda$ in the $(ij)^{th}$ position, and $0$s elsewhere. A matrix that can be written as $E_{ij}(\lambda )$ for some $i,j,\lambda$ is termed an elementary matrix.

Every elementary matrix can be written as a commutator of two invertible matrices in either of these cases:

• $n\geq 3$.
• $k$ has at least three elements.

Related facts

Facts about elementary matrices

• Every elementary matrix is a commutator of elementary matrices: This holds over any unital ring, but requires $n\geq 3$.
• Every elementary matrix is a commutator of unimodular matrices: This requires a somewhat more stringent condition on $k$: $k$ should have more than three elements.
• Every elementary matrix is the commutator of an invertible and an elementary matrix: This is a stronger version of the same statement, and has essentially the same proof.

Facts about the general and special linear groups

• Special linear group is perfect: Holds when $n\geq 3$ and $k$ has at least four elements.
• Commutator subgroup of general linear group is special linear group: Holds when $n\geq 3$ and $k$ has at least three elements.
• Lower central series of general linear group stabilizes at special linear group: Holds when $n\geq 3$ and $k$ has at least three elements.

Facts used

1. Every elementary matrix is a commutator of elementary matrices

Proof

The case $n\geq 3$

This follows directly from fact (1).

The case $n=2$ and $k$ has more than two elements

We need to show that the matrices $E_{12}(\lambda )$ and $E_{21}(\lambda )$ can be expressed as commutators of invertible mtarices. We show this for $E_{12}(\lambda )$. A similar argument works for $E_{21}(\lambda )$.

Pick $\mu \in k$ such that $\mu \neq 0,1$. Consider:

$g={\begin{pmatrix}\mu &0\\0&1\end{pmatrix}},\qquad h={\begin{pmatrix}1&\lambda /(\mu -1)\\0&1\end{pmatrix}}$.

A computation shows that the commutator of $g$ and $h$ is $E_{12}(\lambda )$.