An elementary matrix is obtained by performing a single elementary row operation on the identity matrix.

Three types of elementary matrices:

• Row swap: $E_{ij}$ swaps rows $i$ and $j$
• Row scaling: $E_i(c)$ multiplies row $i$ by nonzero scalar $c$
• Row addition: $E_{ij}(c)$ adds $c$ times row $j$ to row $i$

Key property:

Performing a row operation on $A$ is equivalent to left-multiplying by the corresponding elementary matrix:

$ E_k \cdots E_2 E_1 A = \text{REF}(A) $

Applications:

• Finding matrix inverses: $A^{-1} = E_k \cdots E_2 E_1$ (when $\text{REF}(A) = I$)
• Solving linear systems
• Computing determinants (as product of pivot entries)