Matrix rank calculator

How to use rank of a matrix calculator?

Complete the steps below to find the rank of a matrix by this calculator.

1. Choose the order of the matrix.
2. Input the entries of the matrix.
3. Click calculate.

Now let’s know what the rank of a matrix is and how to find it.

What is the rank of a matrix?

The rank of a matrix represents the number of rows that are unique. In other words, the rows that cannot be expressed in a linear equation with other rows in a dependent relation.

It is called rank and not ROW rank because the row rank of a matrix is equal to its column rank and eventually it is the rank of the matrix. 

The rank of a matrix cannot exceed the minimum rows or columns. If you have a 2 by 4 matrix, then the rank of the matrix will be either 2 or less than 2.

Example of the rank of a matrix:

Consider you have this 3 by 3 matrix.

At first, it may look that this matrix has rank 3. Because all 3 rows look independent and unique. But if we get a closer look, the answer will be different.

See row number two. 

If we compare it to row one, we can see that there is some sort of relevance in both rows. And that relation can be expressed as:

r₂ x 3 = r₁

So that cancels out one row. Now see the third row.

It can be expressed in the linear equation as:

r₃ = 5(r² - r₁)

That means there is only one unique row in this example and hence the rank of this matrix is 1.

But there are two ways to speed up your process of finding the rank. The first one is specifically for square matrices. 

If the determinant of a square matrix is not equal to zero (i.e |A| ≠ 0) then the rank of the matrix is equal to the order of the matrix. For example, look at this 2x2 matrix:

The determinant of this matrix is = |A| = (2)(3)-(1)(1)
                                                             = 6 - 1
                                                             = 5 ≠ 0

This means its rank is 2, the order of the matrix.

Speaking of matrices, care to look at the transpose calculator?    
 
Now, the question arises, “How do you find the rank of a rectangular matrix?”. And also those square matrices which have rank less than their orders. The answer is the “Reduced row echelon form method”.

In this method, you perform row reduction until the matrix is in the echelon form. After that, you see the number of non-zero rows, and that is the rank of the matrix.

What is the echelon form? There are three rules that tell if the matrix is in echelon form.

1. Non-zeros rows above the all-zero rows.
2. The first non-zero digit of the row must be one.
3. The leading 1s must be on the right side of the leading 1 of the row above it.

For example:

All the first non-zero entries are 1 in this example and each succeeding row has its first non-zero entry on the right of the previous one. Lastly, non-zeros are coming first.