Rank of Matrix

The rank of a matrix is a core concept in linear algebra, particularly in engineering mathematics. It serves as a measure of the non-degeneracy of a system of linear equations. Understanding how to calculate the rank of a matrix is crucial for solving various mathematical problems, including those involving systems of linear equations, transformations, and vector spaces.

In this blog, we will explore what the rank of a matrix is, how to find the rank of a matrix, particularly by using the echelon form, and work through some rank of matrix questions with solutions.

1.0What is the Rank of a Matrix?

The rank of a matrix is the maximum number of linearly independent(L.I) row vectors or column vectors within the matrix. It indicates the dimension of the vector space spanned by its rows or columns. In simple terms, the rank gives the number of independent equations in a system of linear equations represented by the matrix.

The matrix A is said to be of rank r, if

1. There is at least one minor of A of order r which does not vanish.
2. Every minor of A of order (r + 1) or higher vanishes.

Rank = Number of non-zero row in upper triangular matrix.

In brief, the rank of a matrix is determined by the largest order of any non-zero minor within the matrix.

From the above definition of the rank of a matrix we have the following two useful results-

1. If a matrix has a non- zero minor of order r, its rank is ≥ r.
2. If all minors of a matrix of order r + 1 are zero, its rank is ≤ r.

The rank of a matrix A shall be denoted by ρ(A). Here 

The rank of a matrix A is denoted by ρ(A). The symbol ρ is a Greek letter pronounced as "rho," so ρ(A) should be read as "rho of A" or "rank of A."

2.0Why is the Rank of a Matrix Important?

In engineering mathematics, the rank of a matrix helps in determining:

• Whether a system of linear equations has a unique solution, infinitely many solutions, or no solution at all.
• The invertibility of a matrix (a square matrix is invertible if and only if its rank is equal to its size).
• The dimension of the image of a linear transformation represented by the matrix.

3.0How to Calculate the Rank of a Matrix?

The rank of a matrix can be determined using three different methods. Among these, the simplest method is converting the matrix into echelon form. The three methods are:

1. Minor Method
2. Echelon Form Method
3. Normal Form Method

1. Minor Method:

Here are the steps to determine the rank of a matrix A using the minor method:

• Calculate the Determinant: If A is a square matrix, first find its determinant. If det(A) ≠ 0, then the rank of A is equal to the order of the matrix.
• Check Minors: If det(A) = 0 (in the case of a square matrix) or if A is a rectangular matrix, look for a non-zero minor of the largest possible order. If such a non-zero minor exists, the rank of A is equal to the order of that minor.
• Reduce and Repeat: If all minors of the largest possible order are zero, reduce the order by one and repeat the process. Continue this until you find a non-zero minor, and the rank of A will be the order of that minor.

2. Echelon Form Method:

A matrix A = (a_ij)_mn is said to be in Echelon form, If

• Every row of A which has all its entries 0 occurs below every row which has a non-zero entry. 
• The number zero preceding the first non-zero element in a row is less than that the number of such zeroes in the succeeding (or next) row.
• The first non-zero element in every row is unity.

The echelon form of a matrix is a form where the matrix has been simplified to a step-like structure, which makes it easier to identify the number of linearly independent rows. To find the rank of a matrix by echelon form, follow these steps:

• Transform the matrix into echelon form: This involves using elementary row operations (such as row swapping, scaling rows, and adding/subtracting multiples of rows) to simplify the matrix into a form where each leading entry (the first non-zero number from the left in a row) is 1 and is to the right of the leading entry in the previous row.
• Count the number of non-zero rows: The count of non-zero rows in the echelon form gives the rank of the matrix.

When a matrix is converted in Echelon form, then the number of non-zero rows of the matrix is known as the rank of the matrix A.

For example, $A=\left[\begin{array}{cccc}1& 2& 1& -1\\ 0& 4& 1& 2\\ 0& 0& 0& 0\end{array}\right]$ is in the echelon matrix.

3. Normal form Method:

Every non-zero matrix [say A = (a_ij)_mn] of rank r, by a sequence of elementary transformations can be reduced to the form.

$\mathrm{A}=\left[\begin{array}{ll}{I}_r& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{I}_r\\ 0\end{array}\right]\left[\begin{array}{ll}{I}_r& 0\end{array}\right]\mathrm{or}\left[I_r\right]$

Where I, is a r × r unit matrix of order r and 0 represented null matrix of any order.

These forms are said to be the normal form of canonical form of the given matrix A. 

4.0Applications of Rank in Engineering Mathematics

The concept of rank is not only theoretical but also has practical applications in various fields, including:

• Control Systems: In control theory, the rank of a matrix helps in determining the controllability and observability of a system.
• Signal Processing: The rank of matrices is used in algorithms for noise reduction and data compression.
• Finite Element Analysis: In structural engineering, the rank of matrices is used to analyze the stiffness and stability of structures.

5.0Solved Example of Rank of a Matrix 

Example 1: Given the matrix $A=\left(\begin{array}{lll}2& 4& 6\\ 1& 2& 3\\ 3& 6& 9\end{array}\right)$ 

Solution: 

Transform the matrix into echelon form.

Perform row operations to simplify the matrix:

2R₂ – R₁ and 3R₃ – R₂

This resuts in:$A^{\prime }=\left(\begin{array}{lll}1& 2& 3\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$

Count the number of non-zero rows.

In this case, there is 1 non-zero row. Therefore, the rank of the matrix A is 1.

Example 2: Find the Rank of the Following$MatrixB=\left(\begin{array}{cccc}1& 2& 1& -1\\ 2& 4& 1& 2\\ 3& 6& 2& 1\end{array}\right)$

Solution:

Transform the matrix into echelon form by performing elementary row operations. $B^{\prime }=\left(\begin{array}{cccc}1& 2& 1& -1\\ 0& 0& -1& 4\\ 0& 0& 0& 0\end{array}\right)$

Count the non-zero rows.

Here, there are 2 non-zero rows, so the rank of the matrix is 2.

Example 3: Obtain the Echelon Form Method Matrix:$A=\left[\begin{array}{lll}2& 4& 1\\ 4& 8& 2\\ 1& 2& 3\end{array}\right]$ and find its matrix.

Solution:

1. Convert the matrix to row echelon form using elementary row operations.

$\left[\begin{array}{lll}2& 4& 1\\ 4& 8& 2\\ 1& 2& 3\end{array}\right]$

R₂ = R₂ –2R₁

$\left[\begin{array}{lll}2& 4& 1\\ 0& 0& 0\\ 1& 2& 3\end{array}\right]{\mathrm{R}}_3={\mathrm{R}}_3-\frac{1}{2}{\mathrm{R}}_1\left[\begin{array}{lll}2& 4& 1\\ 0& 0& 0\\ 0& 0& \frac{5}{2}\end{array}\right]$

2. Identify the number of non-zero rows. Here, there are two non-zero rows.

The rank of the matrix A is 2.

Example 4: Using the Minor Method $B=\left[\begin{array}{lll}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ , find rank of matrix B. 

Solution:

1. Calculate the determinant of B:

det(B) = 1(5  9 – 6 × 8) – 2(4× 9 –6 × 7) + 3(4 × 8 – 5 × 7)

det(B) = 1(45 – 48) – 2(36 –42) + 3(32 – 35)

det(B) = –3 + 12 –9 

det(B) = 0

Since det(B) = 0, the matrix is not of full rank. The rank is less than 3.

2. Check the minors of order 2 (i.e., 2 × 2 submatrices). Let's check the minor formed by the first two rows and the first two columns:

$M=\left[\begin{array}{ll}1& 2\\ 4& 5\end{array}\right]$

Calculate the determinant of M:

det(M) = 1 × 5 – 2 × 4

      = 5 –8

      = –3

Since this minor is non-zero, the rank of the matrix B is 2.

The rank of the matrix B is 2.

Example 5: Using the Normal Form Method Matrix: $C=\left[\begin{array}{cccc}2& 1& 3& 4\\ -1& 2& 0& 1\\ 3& 1& 3& 2\end{array}\right]$ , find rank of matrix C.

Solution:

1. Perform row operations to convert C into its normal form.

Swap Row 1 and Row 2 to bring a 1 to the pivot position:

$\left[\begin{array}{cccc}-1& 2& 0& 1\\ 2& 1& 3& 4\\ 3& 1& 3& 2\end{array}\right]$

R₂ = R₂ + 2R₁, R₃ = R₃ –3R₁

$C=\left[\begin{array}{cccc}-1& 2& 0& 1\\ 0& 5& 3& 6\\ 0& 7& 3& 5\end{array}\right]$

${\mathrm{R}}_3={\mathrm{R}}_3-\frac{7}{5}{\mathrm{R}}_2$

$C=\left[\begin{array}{cccc}-1& 2& 0& 1\\ 0& 5& 3& 6\\ 0& 0& -\frac{6}{5}& -\frac{17}{5}\end{array}\right]$

2. Count the number of non-zero rows. There are three non-zero rows.

The rank of the matrix C is 3.

6.0Practice Questions on Rank of a Matrix 

1. Using Echelon Form Method find the rank of the following matrix A: $A=\left(\begin{array}{llll}1& 2& 3& 4\\ 2& 4& 6& 8\\ 1& 1& 1& 1\end{array}\right)$
2. Using Minor Method determine the rank of the matrix B using the minor method:$B=\left(\begin{array}{cccc}0& 1& 2& 3\\ 4& 5& 6& 7\\ 8& 9& 10& 11\\ 12& 13& 14& 15\end{array}\right)$
3. Using Normal Form Method find the rank of the matrix C using the normal form method: $C=\left(\begin{array}{llll}2& 4& 1& 3\\ 1& 3& 0& 2\\ 3& 7& 1& 5\end{array}\right)$
4. Find the rank of the matrix D where $D=\left(\begin{array}{cccc}1& -1& 2& 1\\ 0& 2& -3& 4\\ 3& -2& 0& 5\end{array}\right).$
5. Determine the rank of the square matrix E: $E=\left(\begin{array}{lll}4& 2& 2\\ 2& 1& 1\\ 6& 3& 3\end{array}\right)$