What is Rank of Matrix?

Rank of a Matrix is a fundamental concept in Linear Algebra, which measures the maximum number of linearly independent rows or columns in any matrix. In other words, it tells you how many of the rows or columns of a matrix are not useful and contribute to the overall information or dimensionality of the matrix. Let’s define the Rank of a Matrix.

Rank of a Matrix Definition

Rank of a matrix is defined as the number of linearly independent rows in a matrix.

It is denoted using ρ(A) where A is any matrix. Thus the number of rows of a matrix is a limit on the rank of the matrix, which means the rank of the matrix cannot exceed the total number of rows in a matrix.

For example, if a matrix is of the order 3×3 then the maximum rank of a matrix can be 3.

Note: If a matrix has all rows with zero elements, then the rank of a matrix is said to be zero.

Nullity of Matrix

In a given matrix, the number of vectors in the null space is called the nullity of the matrix or it can also be defined as the dimension of the null space of the given matrix.

Total columns in a matrix = Rank + Nullity

Read More about Rank Nullity Theorem.

Rank of a Matrix is defined as the dimension of the vector space formed by its columns. Rank of a Matrix is a very important concept in the field of Linear Algebra, as it helps us to know if we can find a solution to the system of equations or not. Rank of a matrix also helps us know the dimensionality of its vector space.

This article explores, the concept of the Rank of a Matrix in detail including its definition, how to calculate the rank of the matrix as well as a nullity and its relation with rank. We will also learn how to solve some problems based on the rank of a matrix. So, let’s start with the definition of the rank of the matrix first.