Matrix Rank Calculator — Step by Step Online

The number of rows:

The number of columns:

Method:

matrixA,0,0,matrixmatrixA,1,0,matrixmatrixA,2,0,matrixmatrixA,3,0,matrix

matrixA,0,1,matrixmatrixA,1,1,matrixmatrixA,2,1,matrixmatrixA,3,1,matrix

matrixA,0,2,matrixmatrixA,1,2,matrixmatrixA,2,2,matrixmatrixA,3,2,matrix

matrixA,0,3,matrixmatrixA,1,3,matrixmatrixA,2,3,matrixmatrixA,3,3,matrix

=Solve

About matrix rank calculator

This is a free online matrix rank calculator with complete, detailed, step-by-step description of solutions, that performs operations with matrices up to 99x99 in size with matrix elements of this type: decimal numbers, fractions, complex numbers, variables.

To start the calculation, you need to first enter the size of the matrix in the input field that you can find from the very top of the screen, also there you can choose the desired method of calculation.

A little below you will find a matrix window in which you need to enter matrix elements using the keyboard. The matrix control panel is also located here, which simplifies work with matrices and contains the following control elements:

The first element allows you to expand the matrix window. This can be especially useful in cases where you need to perform calculations with very large matrices that do not fit completely. If the matrix is still not visible after expanding the window, you can change the scale of the matrix using the + / - buttons;
The second element performs the function of copying the matrix input to the memory buffer. This can be useful in cases where you often use the same matrix for calculations, or if you need to move matrices between operations;
And the last element inserts the previously copied matrix, which allows you to speed up the process of entering the matrix to just a few clicks, instead of doing it manually;

And further down you will find a toolbar that allows you to customize the calculator and make it easier to work with it. It is visually divided into three parts, each of which is responsible for the following functionality:

The first allows you to select the number format when the solution result is displayed. Also, here you can turn off comments to the solution of the problem if you have already understood how to solve this problem, and you use the calculator to speed up or check your own calculations. Or you can turn off the step-by-step solution entirely if you only need the result of the solution;
The second contains buttons that allow you to change the type of the matrix input field, erase its elements or the entire matrix, and the largest button with an equal sign, which will take you to the screen with the solution of the problem. All these buttons are duplicated by keys on the keyboard. To find out which key on the keyboard to press, simply hover over one of the buttons and a tooltip will appear with the name of the key. You can also use the arrow keys on your keyboard to move the cursor between matrix input fields;
And the last one allows you to choose the number of digits after the decimal point for rounding non-integer numbers. Also, here you can immediately see an example of how rounded fractions will look;

What is rank of a matrix?

The rank of a matrix is the number of linearly independent rows or columns in the matrix. The number of linearly independent rows and columns in the matrix is always the same. We can also say that the rank of the matrix is equal to the order of the highest non-zero minor of the matrix. The rank of a matrix can be found for matrices of any size and cannot be greater than the number of rows or columns in the matrix.

How to find the rank of a matrix using elementary transformations(Echelon form)?

Using Gaussian elimination, we can reduce the matrix to row echelon form. After that, we just need to count the number of non-zero rows in the resulting matrix, and this value will be equal to the rank of the original matrix.

Matrix rank example

Write the initial matrix 

:

=

To find the rank of matrix 

, reduce the matrix to row echelon form and count the number of nonzero rows;

Iteration 1

From 

2

th row we subtract 

1

th row, multiplied by 

2

;

From 

3

th row we subtract 

1

th row;

1

0

0

2

0

-2

3

0

-2

4

0

-4

Iteration 2

Exchange rows 

2

th and 

3

th;

1

0

0

2

-2

0

3

-2

0

4

-4

0

Divide 

2

th row by 

-2

;

Rank of matrix

Since the number of nonzero rows is 

2

 it follows that rank(

) =

2

;

Answer

rank(A) =

rank(

) =

2

;

SIZE3×4METHODElementary transformations(Echelon form)

Frequently asked questions

What is the rank of a matrix?

The rank is the number of linearly independent rows, which equals the number of linearly independent columns. It is the count of non-zero rows after reducing the matrix to row-echelon form.

How do you find the rank of a matrix?

Reduce the matrix to row-echelon form with elementary row operations and count the non-zero rows. Alternatively, the rank equals the size of the largest non-zero minor (the bordering-minors method).

Can the rank be larger than the number of rows?

No. The rank of an m×n matrix cannot exceed the smaller of m and n, so it is at most min(m, n).

What does full rank mean?

A matrix has full rank when its rank equals min(rows, columns). A full-rank square matrix is invertible; a rank-deficient square matrix is singular.

Calculation methods

Elementary transformations(Echelon form)
Bordering minors

Sources

Google Play App Store