Home | | Maths 12th Std | Gauss Jordan Method

Definition, Theorem, Formulas, Solved Example Problems | Elementary Transformations of a Matrix - Gauss Jordan Method | 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

Chapter: 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

Gauss Jordan Method

Let A be a non-singular square matrix of order n . Let B be the inverse of A.

Gauss-Jordan Method

Let A be a non-singular square matrix of order n . Let B be the inverse of A.

Then, we have AB = BA = In . By the property of In , we have A = In A = AIn .

Consider the equation A = In A           …………………………………………..(1)

Since A is non-singular, pre-multiplying by a sequence of elementary matrices (row operations) on both sides of (1), A on the left-hand-side of (1) is transformed to the identity matrix In and the same sequence of elementary matrices (row operations) transforms In of the right-hand-side of (1) to a matrix B. So, equation (1) transforms to In = BA. Hence, the inverse of A is B. That is, A1 = B.

Note

If E1 , E2 ,L, Ek are elementary matrices (row operations) such that (Ek L E2 E1 ) A = In ,  then A1 = Ek L E2 E1.

Transforming a non-singular matrix A to the form In by applying elementary row operations, is called Gauss-Jordan method. The steps in finding A1 by Gauss-Jordan method are given below:

Step 1

Augment the identity matrix In on the right-side of A to get the matrix [ A | In ] .

Step 2

Obtain elementary matrices (row operations)  E1 , E2 ,L, Ek such that  (Ek L E2 E1 ) A = In .

Apply  E1 , E2 ,L, Ek on [ A | In ] . Then  [(Ek …… E2 E1 ) A | (Ek ….. E2 E1 ) In]. That is, [In  | A1 ].

 

Example 1.20

Find the inverse of the non-singular matrix A = , by Gauss-Jordan method.

Solution

Applying Gauss-Jordan method, we get


 

Example 1.21

Find the inverse of A =  by Gauss-Jordan method.

Solution

Applying Gauss-Jordan method, we get


 

Tags : Definition, Theorem, Formulas, Solved Example Problems | Elementary Transformations of a Matrix , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Gauss Jordan Method | Definition, Theorem, Formulas, Solved Example Problems | Elementary Transformations of a Matrix

Related Topics

12th Mathematics : UNIT 1 : Applications of Matrices and Determinants


Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright © 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.