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StudyMath
# Matrix Rank

##### This online calculator determines the rank of a given matrix

### This page exists due to effort of the following persons:

**Author**- Timur - Matrix Rank
**Created using the work of**- Timur - Matrix Rank

This online calculator determines the rank of a given matrix. Some theory are below the calculator.

Now, some theory.

A minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns.

The rank of a matrix is the greatest order of any non-zero minor in the matrix (the order of a minor being the size of the square sub-matrix of which it is the determinant).

This minor is called the basis minor, and the columns and rows of this minor are called the basis columns and basis rows. Basis columns and basis rows are linearly independent.

Usage of a rank:

Any n columns are linearly dependent if n is greater than rank.

The columns (or rows) of a matrix are linearly dependent when the number of columns (or rows) is greater than the rank, and are linearly independent when the number of columns (or rows) is equal to the rank.

The maximum number of linearly independent rows equals the maximum number of linearly independent columns.

This calculator uses basis minor method to find out matrix rank.

Basis minor method:

In this method we try to find consequently the non-zero minors of the size 1, 2, …, adding on each

stage just one row and one column to the non-zero minor found on the previous stage.

If there is a minor of the order k that is not equal to 0, but all minors of the order k+1, obtained in this process are equal to 0, then the rank of the matrix is k.

Now, some theory.

A minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns.

The rank of a matrix is the greatest order of any non-zero minor in the matrix (the order of a minor being the size of the square sub-matrix of which it is the determinant).

This minor is called the basis minor, and the columns and rows of this minor are called the basis columns and basis rows. Basis columns and basis rows are linearly independent.

Usage of a rank:

Any n columns are linearly dependent if n is greater than rank.

The columns (or rows) of a matrix are linearly dependent when the number of columns (or rows) is greater than the rank, and are linearly independent when the number of columns (or rows) is equal to the rank.

The maximum number of linearly independent rows equals the maximum number of linearly independent columns.

This calculator uses basis minor method to find out matrix rank.

Basis minor method:

In this method we try to find consequently the non-zero minors of the size 1, 2, …, adding on each

stage just one row and one column to the non-zero minor found on the previous stage.

If there is a minor of the order k that is not equal to 0, but all minors of the order k+1, obtained in this process are equal to 0, then the rank of the matrix is k.

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