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Rank of a Matrix

**Definition:**In a matrix

**A**of order

**(m × n)**if,

**(i)**At least one minor of order

**r**is not zero,

**(ii)**Every minor of order

**(r + 1)**is zero,

Then

**r**is said to be the rank of the given matrix

**A**and is denoted by

**ρ(A)**.

Since every minor of order

**(r + 2)**of

**A**can be expressed as a linear combination of the minors of order

**(r + 1)**of

**A**, therefore, we can say that if

**ρ(A) = r**, then every minor of order

**(r + 2)**will also be zero. Similarly, it can be seen that every minor of order

**(r + 3), (r + 4), …**etc will also be zero. Thus we observe that if

**ρ(A) = r**then

**(i)**At least one minor of order

**r**is not zero,

**(ii)**Every minor of order

**(r + 1)**is zero,

In other words, the rank of a matrix is the largest order of any non-vanishing minor.

**Note 1:**If every minor of order greater than or equal to

**r**is zero, than

**ρ(A) < r**.

**Note 2:**If there exists a non-zero minor of order r, then

**ρ(A) ≥ r**.

**Note 3:**For a non-zero matrix, the least value of its rank is one.

**Note 4:**We agree to define rank of a zero matrix as zero.

**Rank of every non-singular matrix of order**

Note 5:

Note 5:

**n**is

**n**.

**Note 6: ρ(I**, where

_{n}) = n**I**

_{n}is a unit of matrix of order

**n**.

**Note 7:**Obviously,

**ρ(A’) = ρ(A)**and

**ρ(A**.

_{0}) = ρ(A)**Services: -**Rank of a Matrix Homework | Rank of a Matrix Homework Help | Rank of a Matrix Homework Help Services | Live Rank of a Matrix Homework Help | Rank of a Matrix Homework Tutors | Online Rank of a Matrix Homework Help | Rank of a Matrix Tutors | Online Rank of a Matrix Tutors | Rank of a Matrix Homework Services | Rank of a Matrix

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