Elementary Matrix

Definition

A matrix $E$ is called an elementary matrix if it can be obtained from an identity matrix by performing a single elementary row operation.

Example

$$(1)\left[\begin{array}{cc}1& 0\\ 0& -3\end{array}\right](2)\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\end{array}\right](3)\left[\begin{array}{ccc}1& 0& 3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right](4)\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

1. Multiply 2nd row by $-3$
2. Interchange row 2 with row 4
3. Add 3 times row 3 to row 1
4. Multiply row 3 by 1 (Thus $E$ can be an identity matrix)