Rank of Matrix

The rank of matrix in simple words may be explained as the number of non-zero rows or columns of a non-zero matrix. We cannot simply find the non-zero rows simply by looking at the given matrix. Hence, to define the rank of matrix more formally, we must know about the minors of a matrix and linearly independent vectors of a matrix.

Minor of a matrix of any order is the determinant of the square sub-matrix of the given matrix. Let A be an m × n, the determinant of any square sub-matrix of A will be a minor of A. So the highest order of any non-singular minor of a matrix is called the rank of matrix.

The rank of matrix is the number of linearly independent vectors of a given matrix. Let us understand more about the rank matrix and its properties.

What is the Rank of Matrix?

The rank of matrix can be defined in several ways. Let us discuss them in brief:

• Rank of Matrix on the basis of Linear Independent Vectors

The maximum number of linearly independent column or row vectors of matrix is called the rank of matrix. If R₁, R₂, …., R_m are the row vectors of a matrix A or C₁, C₂, …, C_n are column vectors of matrix A such that

Then, R₁, R₂, …., R_m or C₁, C₂, …, C_n are said to be linearly independent if

A₁ R₁ + A₂ R₂ + …..+ A_m R_m = 0 ⇔ A₁ = A₂ = … = A_m = 0 where A₁, A₂, …, A_m are scalars

Or k₁ C₁ + k₂ C₂ + … + k_n C_n = 0 ⇔ k₁ = k₂ = … = k_n = 0 where k₁, k₂, …, k_n are scalars.

• Rank of Matrix on the basis of Minor of Matrix

The highest order of non-zero minor of a matrix is said to be the rank of a matrix. If ‘r’ is the rank of the matrix then atleast one minor of the given matrix is of order r and all other minors of order greater than r is zero.

• Rank of Matrix on the basis on Echelon Form

The number of non-zero rows of a matrix reduced in echelon form is called the rank of the matrix.

A matrix is said to be in echelon form if

i.) there are any zero rows, then they should be placed below the non-zero rows.

ii.) the number of zero in front of any row increases according to the row number.

The non-zero rows of an echelon matrix are that matrix’s linearly independent row vectors.

The rank of a matrix A is denoted by r(A) or ⍴(A).

How to Find Rank of Matrix?

Let us understand how to find rank of matrix by taking an example. Let A be a matrix of order

4 × 4 such that

Method 1:

Reduce the given matrix A in a row reduced echelon form by applying elementary row operations. Given

Step 1: Applying elementary row operations, exchange R₁ with R₂, that is R₁ ↔ R₂

Step 2: Applying R₂ → R₂ + (-2)R₁, R₃ → R₃ + R₁ and R₄ → R₄ + (-1)R₁ we get

Step 3: Applying R₂ ↔ R₃, R₃ → R₃ + (-3)R₂ and R₄ → R₄ + (-5)R₂ we get,

Step 4: Applying R₃ → (-⅛)R₃, R₄ → R₄ + (8)R₃

Therefore, B is the required row reduced Echelon form

Thus, ⍴(A) = 3, since there are 3 non-zero rows.

Method 2:

By finding non-singular minors of the given matrix A.

Given

Clearly, |A| = 0, that is A is singular matrix

But the minor

Hence the highest order of the non-singular minor of A is 3

Therefore, ⍴(A) = 3.

Properties of Rank of Matrix

The following are some important properties of the rank of a matrix.

• Let A be any non-zero matrix of any order and if ⍴(A) < order of A then A is a singular matrix.
• Only the rank of a Null Matrix is zero.
• Rank of an Identity Matrix I is the order of I.
• Rank of matrix A_{m × n} is minimum of m and n.
• If A’ and A* are the transpose and tranjugate of matrix A, then ⍴(A’) = ⍴(A) and

⍴(A*) = ⍴(A).

• Two row equivalent matrices have same rank.
• ⍴(A + B) ≤ ⍴(A) + ⍴(B)
• ⍴(A – B) ≥ ⍴(A) – ⍴(B)
• If A and B are two matrix conformable for matrix multiplication the ⍴(AB) ≤ min{⍴(A), ⍴(B)}.
• If the order of a square matrix A is m, then m – ⍴(A) is the nullity of matrix A.

Related Articles

• Transpose of Matrix
• Determinant of a Matrix
• Matrix Multiplication
• Matrix Operations
• Special Matrices

Solved Examples on Rank of Matrix

Example 1:

Find the rank of the matrix

Solution:

Let A =

Then |A| = 1( 21 – 20) – 2( 14 – 12) + 3( 10 – 9)

= 1 – 4 + 3 = 0

Thus A is a singular matrix.

But

Therefore, ⍴(A) = 2.

Example 2:

Are the rows of the matrix

Solution:

If order of the matrix = rank of matrix, then the row vectors of the matrix are linearly independent.

Let A =

Then |A| = 1 × ( 8 – 9) – 1 × (4 – 6) + 2 × (3 – 4) = -1 + 2 – 2 = -1 ≠ 0

Hence, ⍴(A) = 3 = order of A

Therefore, the rows of the given matrix are linearly independent.