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Matrix Rank Tutorial

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Matrix is an array of numbers arranged in rows and columns of order m x n (m rows and n columns). Every single number present in the matrix is called as the element or the entry.
 
Below is the example of the matrix of order 3x3: 
                       [1 2 3]
                       [4 5 6]
                       [7 8 9]

Rank of Matrix:
The matrix rank is determined by the number of independent rows or columns present in it. A row or a column is considered independent, if it satisfies the below conditions.
1. A row/column should have atleast one non-zero element for it to be ranked.
2. A row/column should not be identical to another row/column.
3. A row/column should not be proportional (multiples) of another row/column.
4. A row/column should not be should not be a linear combination of another row/column.

A row or a column is ranked only if it meets the above conditions. For example, the rank of the below matrix would be 1 as the second row is proportional to the first and the third row does not have a non-zero element.
                       [1 2 3]
                       [2 4 6]       
                       [0 0 0]
How to calculate the rank of a matrix:
In this tutorial, let us find how to calculate the rank of the matrix. It can be calculated using various methods. We will calculate the rank based on Gaussian elimination method.

Gaussian elimination method is used to calculate the matrix rank by converting it into the reduced row echelon form. It can be called as reduced row echelon form, if it satisfies the following conditions.

1. The first element in the first row should be the leading element i.e. 1.
2. The leading element in the columns should be to the right of the previous row's leading element.
3. If there are any rows with all zero elements, it should be below the non-zero element rows.
4. The leading element should be the only non-zero element in every column.

If we follow the above steps, then the matrix would become a triangular one, i.e. the elements below or above the main diagonal would become zero.

Main diagonal, of a square matrix, is a diagonal which runs from top left corner and to the bottom right corner.

However, in this session, we will not consider the last (fourth) point as it would not affect the rank of a matrix.

A matrix can be converted to reduced row echelon form by using elementary operations. In this tutorial, there are three basic elementary operations explained.

1.Interchanging two rows (or columns).
2.Multiplying a row (or a column) with a non-zero number.
3.Multiplying a row (or a column) with a non-zero number and adding the result to another row (or a column).

Now, for example let us calculate the matrix rank.
                       [1 2 3]
                       [4 5 6]
                       [7 8 9]

a)Since the first element of the first row is the leading element, we have satisfied the first condition. If the first element is not 1, then we need to convert the element to 1 by using elementary operations.

b)Now, let us make the first element of the second row zero (to form the triangular matrix). Multiply the first row with -4 and add that to the second row. The result is given below. 
                     [1  2  3]
                     [0 -3 -6]
                     [7  8  9]

c)Similarly let us follow the same step for the third row for finding the rank of a matrix. Multiply the first row with -7 and add that to the third row. Below is the result. 
                    [1  2   3]
                    [0 -3  -6]
                    [0 -6 -12]

d)Now, let us multiply the second row with -2 and add the values to the third row to make the second element of the third row zero. The resulting matrix would look like below.

                     [1  2  3]
                     [0 -3 -6]
                     [0  0  0]

Now, since it has been converted to row echelon form, we can find the rank of matrix.
The matrix rank is 2 as the third row has zero for all the elements.
As we saw in this tutorial, the rank can be found in simple steps using Gaussian Elimination method.



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