Rank of a Matrix

Rank of a matrix is number of linearly independent rows or columns in a matrix.

Geometrically speaking, Rank of matrix is the number of dimension it changes to after linear transformation.


Example 1

\(A = \begin{bmatrix} 24 & 36 \\ 12 & 18 \end{bmatrix} \)

Take a look at 2x2 matrix A above.

The second row is a 1/2 times the first row.

Although there are 2 rows, the matrix rank is 1, because second row don't count because of linear dependency on 1st row.

Example 2

\(M = \begin{bmatrix} 25 & 5 & 15 \\ 15 & 3 & 9 \\ 5 & 1 & 3 \end{bmatrix} \)

The matrix M above has 3 rows and 3 columns (3x3 matrix).

As you can see, the 1 and 2 column are fine. But the 3rd column is a linear combination of 1st column.

3rd column = 1/5 * 1st column

So, the matrix rank is 2 (It will lose 1 dimension after transformation)

Significance of Matrix Rank

Matrix rank is a important concept in linear algebra. It helps to know if we can solve a system of linear equation or not. For example, if the matrix rank is less than the number of variables in equation, the system of equations is incosistent and has no solution.

For example::

3 laptop and 2 mobile cost $4100 2 laptop and 3 mobile cost $3500

Now, we can easily find the cost of laptop and mobile individually because here variables are not dependent on each other (rank is 2)

3 laptop and 2 mobile cost $4100 6 laptop and 4 mobile cost $8200

Here, we can't find the individual cost of laptop and mobile, because second equation here is just first equation multiplied by 2.

So, the rank of matrix is 1 and we can't find the individual cost of laptop and mobile.

Full Rank Matrix

A matrix having the rank as high as it can (equal to the number of it columns) is called full rank matrix. A full rank matrix is a matrix which don't lose any information after transformation. Also there always exists a inverse for a full rank matrix.

Methods to Calculate Rank of a Matrix

There are several methods to calculate the rank of a matrix. Some of them are:

Properties of Rank of a Matrix

Here are some properties of rank of a matrix:

  1. Rank of a matrix is a scalar value
  2. Rank of a matrix is always less than or equal to number of columns in the matrix
  3. For non, singular matrix rank is equal to number of columns in the matrix

Frequently Asked Questions

What is the maximum rank of a matrix?

The maximum rank of a matrix is the maximum number of columns in the matrix. If a matrix has n columns, then the maximum rank of the matrix is n.

Determinant and Rank of a Matrix - Are they related?

Determinant and rank of a matrix are related. They both are related to linear transformation of a matrix. The rank of a matrix gives us information about the matrix which determinant cannot.

Determinant of a matrix tells us the factor of a area (in 2D) or volume (in 3D) increases or decreases after transformation Whereas rank of the matrix tells us how many dimensions the matrix will lose after transformation (if any).

What is the minimum rank of a matrix?

The minimum rank a matrix can have is 0. Having a mimimum rank of 0 means all rows and columns are linearly dependent.

What does it mean for a matrix to be full rank?

A matrix having full rank means that it don't loses any information (dimension reduction) after transformation. A matrix that don't lose dimension after transformation is always invertible (has an inverse).

Is non-zero determinant squared matrix always full rank?

Yes, A non-zero determinant square matrix is always full rank. It is always invertible and don't lose any dimension after transformation

What is the rank of a zero matrix?

The rank of a zero matrix is 0. Because all the rows and columns of a zero matrix are linearly dependent.

What is the rank of a singular matrix?

We can't determine the rank of a singular matrix without looking at it's elements. A singular matrix have a determinant of a 0 (whihc means it loses dimension after transformation), but its not as having a rank of 0.

A singular matrix with n columns can have a rank of 0, 1, 2, ... n-1 depending on the elements of the matrix

Use this matrix rank calculator to calculate the rank of a matrix.

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