Elementary row/column operations on matrices

The elementary row/column operations on matrices are:

1) Multiplying the $i$-th row (column) by a number $\lambda \ne 0$.

$R_i := \lambda \cdot R_i$

2) Adding the $j$-th row/column multiplied by a number $\lambda$ to the $i$-th row/column.

$R_i := R_i + \lambda \cdot R_j$

3) Exchanging the rows/columns $i$ and $j$.

$R = R_i$

$R_i = R_j$

$R_j = R$

The interesting thing is that for square matrices each of these operations can be accomplished by matrix multiplication.

Let us define: 

• $E$ - the identity matrix of order $n \times n$.
• $E_{ij}$ - the square matrix of order $n \times n$ which has an element $1$ at position $(i, j)$ and zeroes at all other positions.
• $A$ - any square matrix of order $n \times n$.

Then one can show that:

(1) Multiplying the i-th row/column of A by the number $\lambda \ne 0$ is accomplished by left/right multiplying A with the matrix $A_i(\lambda) = E + (\lambda-1)E_{ii}$

(2A) Adding the $j$-th row multiplied by a number $\lambda$ to the $i$-th row is accomplished by left multiplying A with the matrix $B_{ij}(\lambda) = E + \lambda E_{ij}$

(2B) Adding the $j$-th column multiplied by a number $\lambda$ to the $i$-th column is accomplished by right multiplying A with the matrix $B_{ji}(\lambda) = E + \lambda E_{ji}$

(3) Exchanging the rows/columns $i$ and $j$ is accomplished by left/right multiplying A with the matrix $C_{ij} = E - E_{ii} - E_{jj} + E_{ij} + E_{ji}$

The matrices $A_i(\lambda), B_{ij}(\lambda), C_{ij}$ are usually called matrices of the elementary transformations.