Rank of a Matrix

Define Rank of a Matrix.

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.

How to Find the Rank of a Matrix?

Rank of matrix can be calculated using various methods such as:

• Minor Method
• Using Echelon Form
• Using Normal Form

What is the Rank of Matrix if Determinant of Matrix is not Equal to Zero?

If determinant of a matrix is zero, then the rank of the matrix is equal to the order of the matrix.

When is a Matrix said to be in Echelon form?

A matrix which is in upper triangular form or in lower triangular form is said to be in echelon form.

What is Normal Form of the Matrix?

A matrix is said to be in normal form if it can be written as [Tex] \begin{bmatrix} I_r & 0\\ 0 & 0\\ \end{bmatrix} [/Tex]where I_r is the identity matrix of the order ‘r’.

What is the Rank of Null Matrix?

Rank of a null matrix is zero.

What is the Rank of an Identity Matrix?

Rank of an identity matrix is equal to the order of the matrix.

What is the Relationship Between Nullity and Rank of a Matrix?

Relationship between nullity and rank of a matrix is:

Total columns in a matrix = Rank + Nullity

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.