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:
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.
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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