## Gauss Jordan Inverse Method

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### Description

Gauss Jordan is a variant of the Gauss elimination method in which row reduction operations are done to find out the inverse of the matrix. Here also the row reduced echelon form of the matrix is formed.

DERIVATION

Then we write the augmented matrix as

Then the first step will be,

Where

And

The second step will be

The third and ultimate step:

### Algorithm

INPUT:

A matrix

OUTPUT:

The inverse of the given matrix

PROCESS:

``````Step 1: [taking the inputs from the user]

Read n [the order of the matrix]

for i = 1 to n repeat
for j = 1 to n repeat
[End of ‘for’ loop]
[End of ‘for’ loop]

Step 2: [Gauss Jordan Inverse]
Set c ← n + 1
for i = 1 to n repeat
for j = n + 1 to 2n repeat
If c = j then
Set a[i][j] ← 1.0
else
Set a[i][j] ← 0.0
[End of ‘if’]
[End of ‘for’ loop]
Set c ← c + 1
[End of ‘for’ loop]

for i = 1 to n repeat
for j = 1 to 2n repeat
Print a[i][j]
[End of ‘for’ loop]
Move to next line
[End of ‘for’ loop]

for k = 1 to n repeat
for j = 1 to 2n repeat
Set a[k][j] ← a[k][j]/a[k][k]
[End of ‘for’ loop]
for i = 1 to n repeat
If i = k then
Continue
[End of ‘if’]
Set r ← a[i][k]

for j = 1 to 2n repeat
Set a[i][j] ← a[i][j] - (a[k][j] * r)
[End of ‘for’ loop]
[End of ‘for’ loop]
for i = 1 to n repeat
for j = 1 to 2n repeat
Print a[i][j]
[End of ‘for’ loop]
Move to the next line
[End of ‘for’ loop]
[End of ‘for’ loop]
[Printing the matrix after inversion]
for i = 1 to n repeat
for j = n + 1 to 2n repeat
Print a[i][j]
[End of ‘fot’ loop]
Move to the next line
[End of ‘for’ loop]

[End of Gauss Jordan Inverse Method]   ``````

### Code

1. It is a stable algorithm when pivoted.

2. It puts a matrix in a reduced row echelon form.

3. For a small system, it is more convenient to use Gauss Jordan Method.