The Rank of a Matrix

The rank of the matrix is the order of the largest non-zero minor in the matrix. For a non-zero matrix B, the rank is ‘r’ if:

• All minors of order r + 1 are zero.
  
   
• There is at least one non-zero minor of order ‘r’.
  
   

Hence, the rank of a matrix A can be written as ρ (A), where ρ (rho) is a Greek letter. So, ρ(A) is read as “rank of A” or “rho of A”.
 

Now let’s learn how to find the rank of a matrix. The rank of a matrix can be found using these methods: 

• Minor method 

• Using echelon form 

• Using normal form 
   

Let’s learn how to find the rank of a matrix using the minor method. In the minor method, we focus on the determinants of the minors. Follow these steps to find the rank of a matrix:

• First, find the determinant of the matrix. Let’s consider A as a square matrix
  If A ≠ 0, then the rank and order of matrix A are the same
  If A = 0, then the rank is determined by the largest order of any minor that is non-zero. 

• If all the minors of the order are zero, we repeat the steps above for the non-zero minor of one order smaller than before. Follow these till you find a non-zero minor. 

While using the minor method, we have to follow several steps. 

• If A is a square matrix, find the determinant of A. 
  When det(A) ≠ 0, the order of the matrix is equal to the rank of A. 
  When det(A) = 0, the highest order of any non-zero minor that can be found inside the matrix is equal to the rank of A.  

• If every minor of the order is zero, repeat the steps. Then, find a non-zero minor of order that is 1 less than the order from the previous step.

Finding det(A), using the 3 × 3 determinant formula: 
   det(A) = a (ei - fh) - b (di - fg) + c (dh -eg)
Substituting the values:    
det(A) = 1 (4 × 6 - 6 × 5) - 2 (2 × 6 - 6 × 4) + 3 (2 × 5 - 4 × 4)
Now, we can calculate: 
    = 1 (4 × 6 - 6 × 5) = 1 (24 - 30) = 1 (-6) = -6
    = - 2 (2 × 6 - 6 × 4) = -2 (-12) = 24
    =  3 (2 × 5 - 4 × 4) = 3 (-6) = -18
     det(A) = −6 + 24 − 18 = 0 
Hence, det(A) = 0, the rank of the matrix is less than 3. 

Check for non-zero minors of order 2. 

 Determinant  = det(A) = ad - bc 
                         =  (1 × 5) - (2 × 4) = 5 - 8 
                         = -3 ≠ 0  
As the result is non-zero, the rank of the matrix is:
     The rank of A (ρ(A)) = 2. 

Using Echelon Form: Identifying a non-zero determinant for finding the rank of a matrix using minors is less efficient for large matrices. We can find the rank of a matrix more easily by using a technique known as the Echelon form. The echelon form is used when the matrix is in the form of an upper or lower triangular matrix. By using the elementary row operations, we can convert a matrix to its Echelon form:

• Interchanging two rows. 

• Multiplying a scalar by a row.
  
   
• Adding a multiple of one row to another row.   

To calculate the rank of a matrix using the Echelon form, we have to follow several steps:

• Using the row/column transformations, convert the matrix into Echelon form. 
  
   
• Then, the number of non-zero rows in the resulting matrix equals the rank of the matrix.  
   

A row in a matrix where at least one element is non-zero is called a non-zero row. 
For example, find the rank of the matrix A = 

Now, we convert the matrix to its Echelon form using the elementary row operations. 
For that, apply the row transformation formula:
     Ri → Ri ​− (k) Rj​
Where Ri = the row to be changed
Rj  = the pivot row.
k = the scalar multiple used to eliminate the entry. 

Applying R2 → R2​−4R1 and R3 → R3 - 7R1, to eliminate 4 and 7 in row 1. 

Next, we will apply R3 → R3 - 2R2, we will get:

A row that has at least one non-zero element is called a non-zero row. In the final matrix, there are 2 non-zero rows. 
Therefore, the rank of A = ρ(A) = 2 

Using Normal Form: The structure of a matrix in normal form is: 
     
Where Ir = the identity matrix of order “r”, and the other values in the matrix will be zero. For a rectangular matrix, A is converted into the standard form using the elementary row transformations and column operations. This method is used to calculate the rank of both rectangular matrices and square matrices.    
For example, find the rank of the matrix A = 

Now we aim to eliminate elements below the first pivot (1) in the first column:
  R2 → R2 - R1
  R3 → R3 - 2R1
  R4 → R4 - 3R1

Hence, the matrix becomes:
    

Next, eliminate elements below and above the pivot in column 2: 
R1 → R1 - 2R2
R4 →R4 - R2

Then, the matrix becomes:

Next, eliminate above the pivot in column 3:
R1 → R1 + R3
R2 → R2 - R3

Then the result is:
 

Now, eliminate the 2 in column 4 to obtain the normal form of the matrix
 C4 → C4 - 2C1
Now, the matrix becomes:
    
This is in the form: 

The identity matrix I3 appears on the top-left side of the matrix, and all the other rows are zero. Therefore, the rank of matrix A is:
   ρ(A) = 3
 

As we learned how to find the rank of a matrix, using the Echelon form and the normal form. We have seen that the number of non-zero rows in the reduced form of the matrix is equal to the rank of the matrix. This is the row rank of the matrix; it is the maximum number of linearly independent rows in the matrix. Whereas the column rank is the number of linearly independent columns. 
. 
Row rank = the number of non-zero rows 
Column rank = maximum number of linearly independent columns
Hence, row rank = column rank. 
 

The properties of the rank of a matrix are used when performing the basic operations like addition, multiplication, and transformations. Here are the key properties of the rank of a matrix:

• The rank of a zero matrix is 0. 

• An identity matrix of order n × n has a rank of n. 

• The rank of a nonsingular matrix A of order n × n is n, that is 
  ρ(A) = n

• The rank of a matrix A in its Echelon form is equal to the number of non-zero rows. 

• The rank of A in its normal form is equal to the order of the identity matrix in it. 

• If A is a singular square matrix of order n × n, then:
  ρ(A) < n

• If A is a rectangular matrix of order m × n, then the rank is:
   ρ(A) ≤ min{m, n}. 

In our daily lives, we can apply the concept of the rank of a matrix to various situations. Here are a few applications:

• In engineering, physics, and economics, professionals use the rank of a matrix to solve complex systems of linear equations. Using this technique, they can easily identify whether a system has a unique solution, no solution, or infinitely many solutions. For example, electronic engineers use the rank to solve for voltages and currents in electrical circuits. 

• In data science and machine learning, the most important variables in a dataset are identified by tools like Principal Component Analysis (PCA) with the help of rank.

• In economics and optimization, the rank of a matrix is used to solve equations involved in resource allocation and financial modeling. For example, to solve optimization issues with constraints and to maximize production or profit, they employ the rank of a matrix to determine the system of linear equations.