Given an 
matrix A,
the largest order of a nonsingular submatrix is called the rank
of A. According to the definition
The rank of a matrix can be evaluated by applying just the same elementary row and column operations as we use to simplify determinants.
- Interchanging two rows or columns.
- Multiplying a row (column) by a nonzero number.
- Multiplying a row (column) by a number and adding it to another one.
Similar operations are said to be elementary transformations of a matrix.
| Theorem |
|
If a matrix
 is obtained from A
by elementary transformations then  . |
By elementary transformations of a matrix we try to obtain as many zeros as possible to reduce the matrix to the echelon form:
