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# Rank of a matrix

Definition.
The rank of a matrix rows (columns) is the maximum number of linearly independent rows (columns) of this matrix.
Theorem.
The rank of a matrix rows is equal to the rank of a matrix columns.
Definition.
The rank of a matrix A is the rank of its rows or columns.
Notation.
Usually, the rank of matrix A is denoted as rank(A) or rang(A)

## How to Find Matrix Rank

Using the properties of the matrix associated with its rank, was received the method of rank calculation which most often used in practice.
Method
The rank of the matrix is equal to the number of non-zero rows after reducing a matrix to row echelon form, using elementary matrix operations with rows and columns.

Example.
Find the rank of a matrix A
 A = 4 2 0 1 2 1 2 3 0 3 10 1 4 2 4 6

Solution:

R1 - 2R2 → R1 (multiply 2 row by -2 and add it to 1 row);R4 - 2R2 → R4 (multiply 2 row by -2 and add it to 4 row): 4 2 0 1 ~ 0 0 -4 -5 ~ 2 1 2 3 2 1 2 3 0 3 10 1 0 3 10 1 4 2 4 6 0 0 0 0

Interchange rows

 ~ 2 1 2 3 0 3 10 1 0 0 -4 -5 0 0 0 0

the resulting matrix there is a matrix in row echelon form, then rank(A) = 3.