Elementary matrix of the first kind

Definition

Let $R$ be a unital ring and $n$ be a natural number. For $i,j\in \{1,2,\dots ,n\}$ with $i\neq j$ and $\lambda \in R$ , the elementary matrix of the first kind $E_{ij}(\lambda )$ is defined as the matrix with $1$ on the diagonal, $\lambda$ in the $(ij)^{th}$ entry, and zeroes elsewhere.

An elementary matrix of the first kind is usually simply termed an elementary matrix, and is also termed a shear matrix.

Facts

Row and column operations

Multiplying a $n\times m$ matrix on the left by a $n\times n$ elementary matrix $E_{ij}(\lambda )$ corresponds to the row operation $R_{i}\mapsto R_{i}+\lambda R_{j}$ . Such a row operation is termed an elementary row operation.

Multiplying a $m\times n$ matrix on the right by a $n\times n$ elementary matrix $E_{ij}(\lambda )$ corresponds to the column operation $C_{i}\mapsto C_{i}+\lambda C_{j}$ . Such a column operation is termed an elementary column operation.

Invertibility and determinant

For $i\neq j$ and for $\lambda ,\mu \in R$ , we have:

$E_{ij}(\lambda )E_{ij}(\mu )=E_{ij}(\lambda +\mu )$ .

From this, we can deduce that the map:

$R\to M_{n}(R)$

given by:

$\lambda \mapsto E_{ij}(\lambda )$

lands inside the group of units $GL_{n}(R)$ of $M_{n}(R)$ , and gives an injective homomorphism from the additive group of $R$ to $GL_{n}(R)$ .

Further, when $R$ is a commutative unital ring, the determinant of $E_{ij}(\lambda )$ is $1$ , so $E_{ij}(\lambda )\in SL_{n}(R)$ .

In fact, we have: