An elementary matrix is expressed by performing a single elementary row operation on an identity matrix.The operation on a row is denoted by the notation R_{i} and column operation is denoted using the notation C_{j} while k represents the scalar quantity. There are three types of elementary row operations:

**1. Interchange two rows or columns of a matrix. R _{i} ↔ R_{j }and C_{i} ↔ C_{j}**

For example, applying R_{1}↔R_{2} to a matrix A = $\lbrack

\begin{matrix}

1 & 4 & \\

9 & 6 & \\

\end{matrix}

\rbrack $, we obtain $\lbrack \begin{matrix}

9 & 6 & \\

1 & 4 & \\

\end{matrix}

\rbrack $

**2. Change a row by adding to it a multiple of another row, R _{i} → R_{i }+ kR_{j} and C_{i} → C_{i} + kC_{j} .**

For example, applying R_{2}R_{2} – 2R_{1}, to P = $

\lbrack \begin{matrix}

1 & 2 & \\

2 & -1 & \\

\end{matrix}

\rbrack $, we obtain $\lbrack \begin{matrix}

1 & 2 & \\

0 & -5 & \\

\end{matrix}

\rbrack $

**3. Multiply each element in a row by the same nonzero scalar number, k. R _{i} → k R_{i }and C_{j} → kC_{j}.**

For example, by applying C_{3}, to B = $

\lbrack \begin{matrix}

1 & 2 & 1 & \\

-1 & 3 & 1 & \\

\end{matrix}

\rbrack $, we obtain$\lbrack \begin{matrix}

1 & 2 & \frac{1}{2} & \\

-1 & 3 & \frac{1}{2} & \\

\end{matrix}

\rbrack $

Likewise, the interchange of two equations, addition or multiplication does not alter the solution set. We can undo these steps by simple operations, for example, we can undo by multiplying the new equation by 1/*c* (since *c ≠* 0), producing the original equation.

**Example: Compute E_{1}A, E_{2}A, and E_{3}A, and show how these products are obtained by elementary row operations on A.**

An interchange of R_{1} and R_{2} of *A* produces *E*_{2}*A* and multiplication of R_{3}5R_{3 }produces *E*_{3}*A*. Left-multiplication by *E*_{1} in Example 1 has the same effect on any 3 X n matrix.Since E_{1}∙ I = E_{1, }we see that *E*_{1} itself is produced by this same row operation on the identity.

Therefore, If an elementary row operation is performed on an m x n matrix *A*, the resulting matrix can be written as *EA*. In this case, the m x m matrix *E* is created by putting the same row operation on *I _{m}*.Each elementary matrix

*E*is invertible. Note that, the inverse of

*E*is the elementary matrix and it is of the same type that transforms

*E*back into

*I*.To calculate the inverse of matrix A, we apply the elementary row operations on the augmented matrix [A I] and reduce this matrix to the form of [I B]. The right-hand side of this augmented matrix B is the inverse of A.

**Example:** Consider the matrix:$A=\lbrack \begin{matrix}

1 & 0 & 2 & 3 & \\

2 & -1 & 3 & 6 & \\

1 & 4 & 4 & 0 & \\

\end{matrix}

\rbrack $ and $E=\lbrack \begin{matrix}

1 & 0 & 0 & \\

0 & 1 & 0 & \\

3 & 0 & 1 & \\

\end{matrix}

\rbrack $.

The elementary matrix provides EA$=\lbrack \begin{matrix}

1 & 0 & 2 & 3 & \\

2 & -1 & 3 & 6 & \\

4 & 4 & 10 & 9 & \\

\end{matrix}

\rbrack $.

This is the same matrix as A that results when we add 3 times the first row of A to the third row.