The Rank of a Matrix

The rank of a matrix is the order of its largest non-zero minor. 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. 
   

The minor method involves the following 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
\(A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 4 & 5 & 6 \end{bmatrix} \)

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 containing 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 
\(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)
 

Step 1:  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: 
\(\begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & -6 & -12 \end{bmatrix} \)
    

Step 2: Next, eliminate elements below and above the pivot in column 2: 

\(R1 → R1 - 2R2\)

\(R4 →R4 - R2\)

Then, the matrix becomes:
\(\begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & 0 & 0 \end{bmatrix} \)

Step 3: Next, eliminate above the pivot in column 3:

\(R1 → R1 + R3\)

\(R2 → R2 - R3\)

 

Step 4: 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:

\(ρ(A) = 3\)

We have 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.

Therefore, the row rank is always equal to the 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}; where m is the number of rows and n is the number of columns.

Mastering the rank of a matrix is essential for solving systems of linear equations and understanding linear algebra concepts. These practical tips will help you build a strong foundation and solve rank problems more efficiently.
 

• Understand the Definition Clearly: Know that the rank of a matrix is the maximum number of linearly independent rows or columns. Understanding this conceptually makes all methods easier to grasp.
   
• Practice Row Reduction (Echelon Forms): Learn how to convert matrices to row-echelon or reduced row-echelon form. Most rank problems are solved efficiently using this method.
   
• Remember Both Row and Column Rank Are Equal: The rank of a matrix is the same whether you calculate it using rows or columns. This is a helpful check when solving problems.
   
• Learn Determinant Tricks: For square matrices, if the determinant is non-zero, the matrix has full rank. This helps quickly identify the rank in special cases.
   
• Identify Linearly Dependent Rows/Columns: Practice spotting rows or columns that are multiples of each other or linear combinations, as these do not increase rank.

The concept of matrix rank has many practical applications. Here are a few real life-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. 
   
• Computer Graphics and Image Processing. Matrix rank helps in image compression and reconstruction. By approximating an image with a lower-rank matrix, storage and transmission become more efficient while preserving most of the important details.
   
• Network Analysis: In network theory, the rank of adjacency matrices is used to analyze connectivity and detect independent paths in communication, transportation, and social networks.