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Rank of the Matrix – Definition, Formulas, Examples

Rank of the 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 matrix is its rank.

The rank is often expressed as rank(A) or rk(A); in some cases, such as rank A, the parentheses are omitted.

The Rank of a Matrix: Definition

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.

Rank of the Matrix 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 Matrix- Formula

  • Minor method
  • Echelon form

Steps to Find the Rank of the 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.

Steps to Find the Rank of the 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.

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FAQs on Rank of a Matrix

Q. What is the meaning of the Matrix?

Ans. A matrix is the environment or context in which anything develops and grows, such as a civilization. A matrix is a set of numbers, symbols, or letters arranged in rows and columns for the purpose of solving mathematical problems.

Q. What is matrix formula used for?

Ans. Matrix is a method of arranging numbers in rows and columns, as well as expressions and symbols. Matrix formulae are used to solve linear equations and other mathematical functions such as calculus, optics, quantum mechanics, and others.

Q. What is a Orthogonal Matrix?

Ans. If the transpose of a square matrix with real numbers or elements equals the inverse matrix, the matrix is said to be orthogonal. To determine whether a given matrix is orthogonal, first determine its transpose. Then, using the transpose, multiply the provided matrix. The provided matrix is orthogonal if the product is an identity matrix; otherwise, it is not.

Q. What is a Symmetric Matrix?

Ans. A symmetric matrix is a square matrix that is identical to its transpose in linear algebra. Only square matrices can be symmetric since equal matrices have equal dimensions. Check to see if the matrix’s transpose is the same as the original matrix. The matrix is symmetric if the transpose matrix and the original matrix are equal.

Q. What is a matrix sheet?

Ans. A matrix diagram is a common lean sigma tool for visualising interactions between two, three, or four groupings of data. The most popular types of templates to utilise as a starting point for creating your own tools from scratch — for a variety of creative reasons — are the matrix and checklist templates.

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