# Rank of a matrix

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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**)

## The properties of the matrix associated with the rank

- The rank of the matrix not change if to its rows (columns) apply elementary matrix operations.
- Rank of matrix in row echelon form is equal to the number of its non-zero rows.

## 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:**

R_{1} - 2R_{2} → R_{1} (multiply 2 row by -2 and add it to 1 row);R_{4} - 2R_{2} → R_{4} (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.

**Answer:** rank(A) = 3.

MatrixMatrix Definition. Main informationSystem of linear equations - matrix formTypes of matricesMatrix scalar multiplicationAddition and subtraction of matricesMatrix multiplicationTranspose matrixElementary matrix operationsDeterminant of a matrixMinors and cofactors of a matrixInverse matrixLinearly dependent and independent rowsRank of a matrix

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