Elementary matrices

Definition: An elementary matrix is a matrix obtained by performing a single elementary row operation on the identity matrix. There are three types, corresponding to the three types of row operations:

1. Row swap $E_{swap}$: swap rows $i$ and $j$ of $I$
2. Row scaling $E_{scale}$: multiply row $i$ of $I$ by nonzero scalar $c$
3. Row replacement $E_{replace}$: add $c$ times row $j$ to row $i$ of $I$

Key property: Left-multiplying a matrix $A$ by an elementary matrix performs the corresponding row operation on $A$:

$$EA = \text{result of applying the row operation to } A$$

Properties:

• Every elementary matrix is invertible
• The inverse of an elementary matrix is also elementary
• Any invertible matrix can be expressed as a product of elementary matrices

Elementary matrices are fundamental in understanding Gaussian elimination and matrix factorization.