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# 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           ÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽÔÇŽ..(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, AÔłĺ1 = B.

### Note

If E1 , E2 ,L, Ek are elementary matrices (row operations) such that (Ek L E2 E1 ) A = In ,  then AÔłĺ1 = 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 AÔłĺ1 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 ÔÇŽÔÇŽ E2 E1 ) A | (Ek ÔÇŽ.. E2 E1 ) In]. That is, [In  | AÔłĺ1 ].

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