Rank of a Matrix - Definition, Formula, Examples and Properties

Matrix Rank is a fundamental concept in linear algebra, representing the maximum number of linearly independent rows or columns in a matrix. It provides valuable insights into the dimensionality and the solvability of linear systems. Understanding the rank of a matrix is essential for various applications in mathematics, engineering, and computer science.

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• 1. Introduction to the Rank of a Matrix
• 2. What is the Rank of a Matrix
• 3. How to Find the Rank of a Matrix
• 4. Finding Rank of a Matrix by Minor Method
• 5. Finding the Rank of a Matrix Using Echelon Form
• 6. Rank of a Matrix Using Normal Form
• 7. Row and Column Rank of a Matrix
• 8. Properties of Rank of a Matrix
• 9. Rank of a Matrix Solved Examples
• 10. Practice Questions on Rank of a Matrix
• 11. FAQs on Rank of a Matrix
• 12. Real-Life Application of Rank of a Matrix
• 13. Conclusion

1. Introduction to the Rank of a Matrix:

Matrix rank is a cornerstone concept in linear algebra, offering a wealth of information about the matrix's properties and the systems of linear equations it represents. Knowing the rank of a matrix helps in understanding the dimensionality and independence of the data it contains.
For example, if we consider the identity matrix of order 3 × 3, all its rows (or columns) are linearly independent; hence, its rank is 3.
Let us learn more about the rank of a matrix along with its mathematical definition, and let us see how to find the rank of the matrix along with examples.

2. What is the Rank of a Matrix:

The rank of a matrix is the maximum number of linearly independent rows or columns, or the rank of a matrix is the order of the highest ordered non-zero minor . In simpler terms, the highest number of vectors can stand alone without being expressed as a combination of others within the matrix.

The rank of a matrix A is denoted by $\rho(A)$.

3. How to Find the Rank of a Matrix:

Finding the rank involves transforming the matrix to identify the maximum number of linearly independent vectors. The there most common methods are:

• Echelon Form Method
• Using Minor Method
• Using Normal Form

We have a designed Rank of a Matrix Calculator , which will give you a step-by-step explanation of the process to obtain the rank of any matrix of any order. Please check out that, too.

4. Finding Rank of a Matrix by Minor Method:

The minor method involves determining the largest non-zero determinant of any square submatrix (minor) within the matrix.

Identify the Minors of the given Matrix:

• Start with the highest-order minors (determinants of the largest possible square submatrices).
• Calculate the determinant of these submatrices.

Check the value of Determinants:

• If you find a non-zero determinant for a submatrix of order n, the rank is at least n.
• Continue this process down to the smallest minors.

Now, determine the Rank:

• The rank is the order of the largest non-zero minor.

Example: Find the rank of a matrix $\rho(A)$ if A = $\begin{bmatrix} 3 & 2 & 1 \\ 5 & 1 & 7 \\ 3 & 2 & 1 \end{bmatrix}$
Solution Det.(A) = -39 32 7 = 0
So $\rho(A) \ne$ Order of the Matrix means $\rho(A) \ne 3$
Now, we will see whether we can find any non-zero minor of order 2.
$\begin{vmatrix} 3 & 2 \\ 5 & 1 \end{vmatrix}$ = 3 -10 = -7 $\ne$ 0
So, there exists a minor of order 2, which is non-zero. So, the rank of A, $\rho(A)$ = 2

5. Finding the Rank of a Matrix Using Echelon Form:

The echelon form method involves transforming the matrix into its row echelon form (REF) or reduced row echelon form (RREF) through row operations.

Reduce the matrix to Row Echelon Form:

• To transform the matrix into REF, Use elementary row operations (swap, scale, and add/subtract rows).
• REF has zeros below each leading entry (pivot).

Count Non-Zero Rows:

• The number of non-zero rows in REF is the rank of the matrix.

Example: Find the rank of the given matrix A = $\begin{bmatrix} 1 & 5 & 3 \\ 4 & 2 & 0 \\ 2 & 3 & 4 \end{bmatrix}$
Solution:
Applying Row operation to reduce the matrix in REF
$R_2 \to (R_2 - 4R_1)$ and $R_3 \to (R_3 - 2R_1)$
Obtained matrix = $\begin{bmatrix} 1 & 5 & 3 \\ 0 & -18 & -12 \\ 0 & -7 & -2 \end{bmatrix}$
Now, again, applying Row operation in Row 3
$R_3 \to (R_3 - \frac{7}{18}R_2)$ Obtained matrix = $\begin{bmatrix} 1 & 5 & 3 \\ 0 & -18 & -12 \\ 0 & 0 & \frac{8}{3} \end{bmatrix}$
So, the number of non-zero rows = 3
Hence, Rank of the matrix $\rho(A)$ = 3

6. The rank of a Matrix Using Normal Form

Normal or canonical forms simplify the matrix to make the rank more apparent.

Transform Matrix:

• Use row and column operations to convert the matrix to a simpler form.

Identify Leading Entries:

• The number of leading entries (non-zero elements in each row) determines the rank.

Example: Find the rank of the given matrix A = $\begin{bmatrix} 1 & 2 & 2 & 1 \\ 2 & 1 & 0 & 1 \\ 2 & 2 & 4 & 2 \\ 4 & 2 & 1 & 1 \end{bmatrix}$
Solution:
Applying Row operation to reduce the matrix in REF
Use our RREF calculator to convert A into RREF.
Obtained matrix = $\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$
We can write the above matrix as = $\begin{bmatrix} I_4 & 0 \\ 0 & 0 \end{bmatrix}$
So, the number of non-zero rows = 4
Hence, Rank of the matrix $\rho(A)$ = 4

7. Row and Column Rank of a Matrix:

A matrix's row rank and column rank are always equal and represent the number of linearly independent rows or columns.

Example: A = $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
Number of non-zero Rows = 3
Number of non-zero Columns = 3
then, The row and column ranks of this 3x3 identity matrix are 3.

8. Properties of Rank of a Matrix:

• Row and Column Equivalence: Row rank is always equal to Column Rank.

• Invariant Under Transformations: Elementary row and column operations do not change the rank

• Submatrix Relationship: The rank of a matrix is at least as large as the rank of any of its submatrices.

• Addition Property: rank(AB) $\le$ rank(A) rank(B)

• Multiplication Property rank(AB) $\le$ min(rank(A), rank(B))

• The rank of a zero matrix is 0.

• The rank of an identity matrix of order n is n itself.

• If A is a nonsingular matrix of order n, then its rank is n. i.e., $\rho(A) = n$

• If A is in normal form, then the rank of A = the order of the identity matrix.

• If A is in Echelon form, then the rank of A = the number of non-zero rows of A.

• If A is a singular matrix of order n, then $\rho(A) < n$

• If A is a rectangular matrix of order m x n, then $\rho(A) \le min(m, n)$

9. Rank of a Matrix Solved Examples:

Example 1: Find the Rank of $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ using REF method.
Solution:
Using Row Operation $R_2 \to (R_2 - 3R_1)$
Obtained REF Matrix = $\begin{bmatrix} 1 & 2 \\ 0 & -2 \end{bmatrix}$
Total number of non-zero Rows = 2
then, Rank of matrix = 2

Example 2: Find the Rank of $\begin{bmatrix} 1 & 2 & 1 \\ 3 & 0 & 3 \\ 6 & 0 & 6 \end{bmatrix}$ using RREF method.
Solution:
Using Row Operation
$R_2 \to (R_2 - 3R_1)$ and $R_3 \to (R_3 - 2R_2)$
Obtained Matrix = $\begin{bmatrix} 1 & 2 & 1 \\ 0 & -6 & 0 \\ 0 & 0 & 0 \end{bmatrix}$
Total number of non-zero Rows = 2
then, Rank of matrix = 2

10. Practice Questions on Rank of a Matrix:

Question 1: Find the rank of $\begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$
Options:
(A) 0
(B) 1
(C) 2
(D) 3

Question 2: Find the rank of $\begin{bmatrix} 2 & -1 & 7 \\ 6 & 8 & 3 \\ -7 & 5 & 4 \end{bmatrix}$
Options:
(A) 0
(B) 1
(C) 2
(D) 3

11. FAQs on Rank of a Matrix:

What do you mean by the rank of the matrix?

The rank of a matrix is the maximum number of linearly independent rows or columns it contains.

Can the Rank of a Matrix be Zero?

Yes, if all elements of the matrix are zero, its rank is zero.

How to Find the Rank of the Matrix?

The rank can be found using methods such as row echelon form, minor method, and normal form.

Can the rank of a matrix exceed the number of rows or columns?

No, the rank cannot exceed the smaller of the number of rows or columns.

What is the Rank of a Matrix of Order 3 × 3?

The rank can be at most 3, depending on the linear independence of its rows or columns.

What do you mean by rank zero matrices?

Rank zero matrices have all elements equal to zero.

What is the Rank of a Matrix of Order 2 × 2?

The rank can be at most 2, depending on the matrix's entries.

What is the Nullity of a Zero Matrix?

The nullity equals the number of columns in the zero matrix.

How to Find the Rank of a Matrix Using Determinant?

Identify the largest non-zero determinant of any square submatrix.

Where can I get the methods to find Matrix Rank?

Methods are detailed in linear algebra textbooks and online educational resources.

What is the Rank of a Null Matrix?

The rank of a null matrix is zero.

What is the use of Matrix rank?

Matrix rank helps determine the solvability of linear systems and the dimension of vector spaces.

What is the Shortcut to Find the Rank of a Matrix?

Reduce the matrix to echelon form and count the non-zero rows.

What are the applications for the rank of Matrix?

Applications include signal processing, data compression, and system analysis.

What Does the Rank of a Matrix Tell Us?

It indicates the number of linearly independent rows or columns.

Can the Rank of a Matrix Ever be Greater than the Number of Rows or Columns?

No, it cannot be greater.

What is the Relation Between the Rank of a Matrix and Eigenvalues?

The rank can provide insights into the number of non-zero eigenvalues.

12. Real-Life Application of Rank of a Matrix:

Matrix rank is pivotal in fields like data science for dimensionality reduction, engineering for system analysis, and computer graphics for transformations.

13. Conclusion:

Grasping the concept of matrix rank equips you with tools to analyze and solve complex linear systems whether using the minor method or echelon form, determining the rank is essential for understanding the matrix's structure and applications in various scientific and engineering disciplines.

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