elementary matrix

An elementary matrix is the identity matrix with one row operation applied. Left-multiplying a matrix $A$ by an elementary matrix performs that row operation on $A$ (right-multiplying performs a column operation).

Three types:

$$\underset{\text{swap rows 2,3}}{\underbrace{\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)}}\underset{\text{scale row 2 by }\lambda }{\underbrace{\left(\begin{array}{ccc}1& 0& 0\\ 0& \lambda & 0\\ 0& 0& 1\end{array}\right)}}\underset{\text{add }\lambda \times \text{row 1 to row 2}}{\underbrace{\left(\begin{array}{ccc}1& 0& 0\\ \lambda & 1& 0\\ 0& 0& 1\end{array}\right)}}$$

Row swap: left vs right multiplication

$$E=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right),A=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$$

Left-multiplying ($EA$) swaps rows 2 and 3:

$$EA=\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{31}& a_{32}& a_{33}\\ a_{21}& a_{22}& a_{23}\end{array}\right)$$

Right-multiplying ($AE$) swaps columns 2 and 3:

$$AE=\left(\begin{array}{ccc}{a}_{11}& a_{13}& a_{12}\\ a_{21}& a_{23}& a_{22}\\ a_{31}& a_{33}& a_{32}\end{array}\right)$$

Every invertible matrix can be written as a product of elementary matrices.
This is how gaussian elimination works: $EA=U$ where $E=E_k\cdots E_2{E}_1$ is a product of elementary matrices and $U$ is upper triangular.

Determinants of elementary matrices: $-1$ for swaps, $\lambda$ for scaling, $1$ for row addition.

If you prove a property for elementary matrices, and the property is preserved under multiplication, it holds for all invertible matrices.