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 $$i \ne j$$ 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.

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 $$i \ne j$$ 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:


 * Elementary matrices generate the special linear group over a field.
 * Elementary matrices generate the special linear group over a local ring
 * Elementary matrices generate the special linear group over a Euclidean ring