A matrix is said to be of rank r when it contains at least one non-zero minor of order r and no such minor of order r + 1. The rank of a matrix is denoted by $\rho$(A).
         By means of elementary transformations every non-zero matrix of rank r can be reduced to one of the following forms

$\left(\mathrm{A}\right)\left[\begin{array}{ccc}{\mathrm{I}}_{\mathrm{r}}& ⋮& 0\\ \dots & \dots & \dots \\ 0& & 0\end{array}\right]$     $\left(B\right) \left[\begin{array}{c}{I}_r\\ \cdots \\ 0\end{array}\right]$     $\left(\mathrm{C}\right) [{\mathrm{I}}_{\mathrm{r}}⋮0]$      $\left(\mathrm{D}\right) \left[{\mathrm{I}}_{\mathrm{r}}\right]$

Where I_r is a r-rowed unit matrix. These are called the normal forms of the given matrix and the value of r is the rank of the 
matrix. In order to find the rank of a given matrix, reduce the matrix to its normal form. This process of reducing a matrix of 
rank r to its normal form is known as ‘The sweep-out process’.