Enter the rank of the matrix into the calculator to determine the maximum number of linearly independent rows or columns, representing the dimension of the vector space spanned by the rows or columns of the matrix; this calculator can also evaluate the rank given the matrix.

## Matrices Rank Formula

The following formula is used to calculate the rank of a matrix:

Rank = r

Variables:

- Rank is the rank of the matrix

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by the rows or columns of the matrix. To calculate the rank of a matrix, perform row operations to reduce the matrix to its row echelon form or reduced row echelon form. The number of non-zero rows in the resulting matrix is the rank of the original matrix.

## What is a Matrices Rank?

The rank of a matrix is a fundamental concept in linear algebra that refers to the maximum number of linearly independent rows or columns in the matrix. It is a measure of the "non-degenerateness" of the system of linear equations represented by the matrix, and it provides insights into the solvability of the system. The rank can be determined through a process called Gaussian elimination, where the matrix is simplified into row echelon form or reduced row echelon form. The rank of a matrix is equal to the number of non-zero rows in its row-echelon form. It is also equal to the dimension of the column space or the row space. The rank gives us valuable information about the matrix, such as whether it is invertible (if the rank equals the number of its columns or rows, the matrix is invertible), and it is used in various fields, including computer graphics, machine learning, and data science.

## How to Calculate Matrices Rank?

The following steps outline how to calculate the Rank of a Matrix.

- First, write down the matrix.
- Next, perform row operations to simplify the matrix.
- Next, count the number of non-zero rows in the simplified matrix.
- Finally, the number of non-zero rows is the rank of the matrix.
- After performing the row operations and counting the non-zero rows, check your answer with the calculator above.

**Example Problem : **

Use the following matrix as an example problem to test your knowledge.

Matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]