Elementary Matrices

An elementary matrix $E$ is a matrix obtained by performing one elementary row operation on an identity matrix.

Example

multiplying row 3 by 4

$$E=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 4\end{array}\right]$$

adding $-2R_3$ to $R_1$

Connection to Inverse

The inverse of an elementary matrix is the opposite row operation!

Applications

Elementary matrices can be used to perform matrix row reduction by multiplying the target matrix by the appropriate elementary matrices.

In other rows, $E\cdot A$ is the result of performing the same elementary row operation on $A$.

To perform several row operations, resulting with a matrix $A_r$:

$$A_r=E_r\cdot E_{r-1}\cdots E_2\cdot E_1\cdot A$$

note the reverse order

If multiplied on the right then the operation is instead performed on the column.

e.g.

$$E=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 3& 1\end{array}\right]$$

$E\cdot A$ multiplies row 3 by 3 and adds it to row 2 of $A$.

$A\cdot E$ multiplies column 2 by 3 and adds it to column 3 of $A$.

Properties

An elementary matrix is always invertible as it is row equivalent to the identity matrix.