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Table of contents
1.
Introduction 
2.
Rank of a Matrix
2.1.
Properties
3.
Determining the Rank of a Matrix
4.
FAQs
5.
Key Takeaways
Last Updated: Mar 27, 2024

Rank of Matrix

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Introduction 

Let us first understand what a matrix is. Oh, I am pretty sure you must be familiar with the matrix in Hollywood, but what does it mean in Mathematics? A matrix (plural: matrices) is a rectangular array or table of numbers(or symbols or expressions) arranged in rows and columns to represent a mathematical object or one of its properties.  A matrix is usually denoted by a capital letter.

Below is an example of a matrix with three rows and three columns.

Rank of a Matrix

The maximum number of linearly independent rows or columns of a matrix is called the rank of the matrix. Let us take matrix A. The rank of matrix A is denoted by ⍴(A). A row or column is linearly independent only if it cannot be represented as a scalar multiple of another row or column or a linear combination of other rows or columns.

If there is a Matrix A with m rows and n columns, the maximum rank of that matrix can bem (if m<=n) or n (if n<m). The rank cannot exceed the number of rows and columns of the matrix. 

Maximum Rank of a matrix Am*n = ⍴(A) = minimum(m,n).

In layman language, the Rank of a Matrix is the number of unique rows (or columns) that are not made up of other rows(or columns).

When all the elements of a matrix become 0, it is said to be of rank 0. The rank of the null matrix is zero since it has no non-zero rows or columns.

Properties

  1. The rank of a matrix of order m×n is ρ(A ) ≤ min{m, n} = minimum(m, n).
  2. A unit matrix of order m has a rank m. 
  3. If a matrix A has order n×n and |A| ≠ 0, its rank will be n.
  4. If a matrix A has order n×n and |A| = 0, A's rank will be less than n.
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Determining the Rank of a Matrix

To determine the rank of a matrix, we should transform it into its echelon form. There are some essential points to consider, such as-

  •  The first element of every non-zero row should be 1.
  • The row having every element as zero should be below the non-zero rows.
  • The total number of zeroes in the next non-zero row should be more than the previous non-zero row.
  • The first non-zero element in any row i of A occurs in the jth column of A, then all other elements in the jth column of A below the first non-zero element of row i are zeros.
  • The first non-zero entry in the ith row of A lies to the left of the first non-zero entry in ( i + 1)th row of A.

A given matrix can be transformed into its echelon form through the following steps:

  1. Interchange two rows (Ri ↔ Rj).
  2. Multiply a row by a non-zero constant, (Ri ↔ kRi) where k ≠ 0.
  3. Add a constant multiple of another row to the given row (Rᵢ ⟶ Rᵢ + kRⱼ), where i ≠ j.

When the matrix is in its echelon form, count the number of non-zero rows. It gives the rank of the matrix.

Check out this problem - Matrix Median

FAQs

  1. How can you determine the rank of a matrix?
    The given matrix is first converted into its row-echelon form. Then the number of non-zero rows gives the rank of the matrix.
  2. Can the rank of a matrix be zero?
    Yes, the rank of a matrix can be zero when all of its elements are zero. A null matrix has rank zero.

Key Takeaways

In this article, we have extensively discussed the concepts of matrices, echelon form and the rank of a matrix. We hope that this blog has helped you enhance your knowledge, and if you wish to learn more, check out our  Coding Ninjas Blog site and visit our Library. Do upvote our blog to help other ninjas grow.

Happy Learning! 

 

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