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Matrices (plural of a matrix) are basic mathematical structures that are employed in many disciplines, including statistics, computer science, physics, engineering, and linear algebra. They offer a succinct and effective means of manipulating and displaying data.

Among the various characteristics of matrices, rank of a matrix is one of the most important. It is because many essential information about a matrix’s structure, capacity to solve linear equation systems, and range of applications can be gleaned from its rank.

## Rank of a Matrix

The rank of a matrix A is the dimension of the vector space formed (or spanned) by its columns in linear algebra. This is the maximum number of linearly independent columns in column A. This is the same as the dimension of the vector space traversed by its rows.

As a result, rank is a measure of the system of linear equations and linear transformation encoded by A’s “nondegeneness.” There are several definitions of rank that are interchangeable. One of the most fundamental aspects of a rank of a matrix is its rank. The rank of a matrix is often expressed as rank(A) or rk(A); in some cases, such as rank A, the parentheses are omitted.

## What is Rank of a Matrix

The number of linearly independent rows or columns in the matrix is referred to as the matrix’s rank. The rank of matrix A is denoted by (A). When all of the elements in a matrix become 0, it is said to be of rank zero. The dimension of the vector space obtained by the matrix’s columns is its rank. A matrix’s rank cannot be more than the number of rows or columns. The null matrix has a rank of zero.

## Define Rank of Matrix

The rank of a matrix is a fundamental concept in linear algebra and refers to the maximum number of linearly independent rows or columns in a matrix. It provides information about the dimensionality of the vector space spanned by the rows or columns of the matrix. The rank of matrix can be calculated using various methods, including row reduction (Gaussian elimination) or by computing the determinant of certain submatrices.

Here are some key points about matrix rank:

**Row Rank and Column Rank:** A matrix can have both a row rank and a column rank. The row rank is the maximum number of linearly independent rows, while the column rank is the maximum number of linearly independent columns. In most cases, the row rank and column rank are equal, but there are exceptions when dealing with non-square matrices.

**Full Rank Matrix:** A matrix is said to be of full rank if both its row rank and column rank are equal to the smaller of the two dimensions (i.e., for an m×n matrix, if rank = min(m, n)). In this case, the matrix is non-singular (has an inverse), and its determinant is non-zero.

**Rank-Deficient Matrix:** If the row rank and column rank are less than the smaller of the two dimensions, the matrix is said to be rank-deficient. Rank-deficient matrices have linearly dependent rows or columns and may not have a unique solution when used in linear systems of equations.

**Rank and Solutions to Linear Systems:** The rank of a matrix plays a crucial role in solving systems of linear equations. If the rank of the coefficient matrix is equal to the rank of the augmented matrix (coefficient matrix combined with the constant vector), then the system has a unique solution. If the ranks are unequal, the system is either inconsistent (no solution) or has infinitely many solutions.

**Rank and Matrix Operations:** The rank of a matrix is preserved under certain matrix operations. For example, adding a multiple of one row to another row or multiplying a row by a non-zero scalar does not change the rank. However, swapping rows can change the rank.

## Rank of Matrix Example

Think of a matrix as a table of numbers, like a spreadsheet. The rank of the matrix tells us how many “important” rows or columns it has. In other words, it helps us understand the dimensionality or the number of independent directions that the data in the matrix spans.

Here’s a Rank of Matrix example:

Imagine you have a 3×3 matrix like this:

`1 2 3`

`4 5 6`

`7 8 9`

To find its rank, you can perform operations on it to simplify it. In this case, you might notice that the third row (7 8 9) is just a multiple of the first row (1 2 3) times 7. So, it doesn’t provide any new information. We can essentially ignore it. This means the rank of this matrix is 2, because you have two independent rows (1 2 3 and 4 5 6) that contain the essential information.

In general, the rank of a matrix helps us understand how many linearly independent rows or columns it has. It’s a crucial concept in linear algebra and has applications in various fields like engineering, computer science, and statistics.

## Rank of Matrix

The rank of Matrix is a fundamental concept in linear algebra that describes the maximum number of linearly independent rows or columns in the matrix. The rank of matrix can be defined for both square and rectangular matrices. Here are some key points about the rank of Matrix:

**Rank of Square Matrix:**

For a square matrix (i.e., a matrix with the same number of rows and columns), the rank is the highest number of linearly independent rows or columns.

If a square matrix has full rank, its determinant is non-zero, and it is invertible (i.e., it has an inverse).

The rank of a square matrix can also be determined by finding the order of the largest non-zero determinant of its submatrices.

**Rank of Rectangular Matrix:**

For a rectangular matrix (i.e., a matrix with a different number of rows and columns), the rank is the maximum number of linearly independent rows or columns. The rank of a rectangular matrix can be found using techniques like Gaussian elimination to row reduce the matrix to its row-echelon form (also known as reduced row-echelon form or RREF).

The number of non-zero rows in the RREF is equal to the rank of the matrix. The rank of a rectangular matrix is always less than or equal to the minimum of the number of rows and columns.

**Rank-Nullity Theorem:**

The rank of a matrix and its nullity (the dimension of its null space) together add up to the total number of columns in the matrix. This is known as the rank-nullity theorem.

**Mathematically, if A is an m x n matrix:**

Rank(A) + Nullity(A) = n, where Rank(A) is the rank of matrix A, Nullity(A) is the nullity (dimension of the null space) of matrix A, and n is the number of columns in A.

## Rank of the Matrix in Detail

The rank of a matrix refers to the maximum number of linearly independent rows or columns in that matrix. In other words, it represents the dimension of the vector space spanned by the rows or columns of the matrix. To determine the rank of a matrix, various methods can be employed, including row reduction (also known as Gaussian elimination) or the use of determinant properties.

The rank provides important information about the properties and solutions of a matrix, such as whether the matrix is invertible (non-singular) or singular, and it is utilized in applications such as solving systems of linear equations, calculating matrix inverses, and determining the dimensions of the column space and null space.

The rank of a matrix can be denoted by “r” and is typically expressed as a non-negative integer value. It satisfies the following properties:

- The rank of the matrix is less than or equal to the minimum of the number of rows and columns.
- The rank of a matrix plus the dimension of its null space (also known as the nullity) equals the number of columns.

By determining the rank of the matrix, one can gain insights into its structure, relationships between its rows and columns, and its ability to represent linear transformations and solve systems of linear equations.

## Matrix Rank 4×4

**Nullity of a Matrix: **The number of vectors in a matrix’s null space is defined as its nullity. In other words, the nullity of A can be defined as the dimension of the null space of matrix A. The total number of columns in matrix A is Rank + Nullity.

A = 1 0 1 0

0 1 0 1

1 1 0 0

1 1 1 1

## Rank of a Matrix 3×3 and 2×2

A matrix with 3 rows and 3 columns are known as 3×3 matrix while a matrix with 2 rows and 2 columns are called 2×2 matrix. We can find the rank of 2×2 and 3×3 matrix just like a 4×4 matrix. That is, either by using Minor Method or Echelon Form, we can determine the rank of a 2×2 and 3×3 matrix.

## Properties of Rank of a Matrix

The matrices’ rank exhibits certain unique properties. These properties are helpful in determining the characteristic of the matrix. Some of the properties of a matrix rank is given below.

- A matrix rank can never be greater than its number of rows or number of columns
- If the matrix is single, its rank is less than its order.
- Identity matrix rank and identity matrix order are equal.
- A zero or null matrix has a rank of zero.
- If a matrix is in normal form, its rank equals the order of the identity matrix contained within it.
- If a matrix is in echelon form, its rank is equal to the total number of non-zero rows.
- If a matrix is not unique, then its rank is equivalent to its order.
- If the matrix is rectangular and has order m x n, its rank is less than {m, n}.

## How to Find Rank of a Matrix?

Determining the rank of a matrix is important in various applications, such as solving systems of linear equations, finding the dimension of the column space (range) of a matrix, and understanding the linear transformations associated with matrices.

Calculating the rank of a matrix is an important step in various applications, including solving linear systems, finding the basis of a vector space, and determining the existence of a unique solution to a system of equations. It can be computed using various algorithms, such as Gaussian elimination or the Singular Value Decomposition (SVD). In mathematics, the two famous methods to find the rank of a matrix is shown hereunder.

- Minor method
- Echelon form

## How to Find Rank of a Matrix by Minor Method

(i) If a matrix has at least one non-zero member, then (A) 1 is true.

(ii) The identity matrix In has a rank of n.

(iii) If matrix A’s rank is r, there must be at least one minor of order r that does not vanish. Every minor of order (r + 1) and higher-order (if any) of matrix A vanishes.

(iv) If A is a m n matrix, then (A) min m, n (v) If and only if (A) = n, a square matrix A of order n must invert.

(v) A square matrix A of order n has to inverse if and only if ρ(A) = n.

## How to Find Rank of a Matrix by Echelon Form

(i) Every non-zero row’s initial element should be 1.

(ii) If every element in a row is zero, that row should be placed below the non-zero rows.

(iii) The total number of zeroes in the next non-zero row should be greater than the previous non-zero row’s total number of zeroes.

We may easily convert the provided matrix to echelon form using simple techniques.

## Rank of a Matrix Solved Examples

**Example:** Find the rank of the matrix A =

by converting it into Echelon form.

**Solution: **The given matrix is

We have to solve this matrix using the Echelon form

Applying R_{2} → R_{2} – 4R_{1} and R_{3} → R_{3} – 7R_{1}, we get:

$\left[\begin{array}{lll}1& 2& 3\\ 0& -3& -6\\ 0& -6& -12\end{array}\right]$

Now, we apply R_{3} → R_{3} – 2R_{2}, we get:

$\left[\begin{array}{lll}1& 2& 3\\ 0& -3& -6\\ 0& 0& 0\end{array}\right]$

As one of the row is all 0, so, it is in Echelon form and so now we have to count the number of non-zero rows.

The number of non-zero rows = 2 = rank of A.

Therefore, ρ (A) = 2.

**Example: **Find the rank of the matrix A =

by converting it into normal form.

**Solution: **We will be solving this matrix by using the normal form method.

o=On applying R_{2} → R_{2} – R_{1}, R_{3} → R_{3} – 2R_{1}, and R_{4} → R_{4} – 3R_{1} we get:

$\left[\begin{array}{llll}1& 2& 1& 2\\ 0& 1& 1& 0\\ 0& 0& 1& 0\\ 0& 1& 1& 0\end{array}\right]$

Now applying, R_{1} → R_{1} – 2R_{2} and R_{4} → R_{4} – R_{2}, we get

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

Apply R_{1} → R_{1} + R_{3} and R_{2} → R_{2} – R_{3},

$\left[\begin{array}{llll}1& 0& 0& 2\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$

Now applying C_{4} → C_{4} – 2C_{1},

$\left[\begin{array}{llll}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$

The above matrix is same as

$\left[\begin{array}{ll}{I}_{3}& 0\\ \\ 0& 0\end{array}\right]$.

Therefore, the rank of A is, ρ (A) = 3.

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