[Linear_Algebra] Gauss Elimination & Gauss-Jordan Elimination

Today, we are going to talk about Gauss Elimination and Gauss–Jordan Elimination, which are crucial for solving linear equations. We discussed Gauss Elimination in the previous post. 

First of all, when GE is performed, the matrix is converted into Row Echelon Form. The next step is back substitution, which means going upward in the matrix to solve for the variables.

 

On the other hand, when GJ is performed, the matrix is converted into Reduced Row Echelon Form, where all the non-diagonal elements are zero and the diagonal elements are 1.

 

So why do we need this? Let's dive into an example.

 

For example, suppose we want to find the inverse matrix of A. 

In this case, we use the equation: A A-1 = I 

The idea is to use Gauss–Jordan elimination to transform A into the Identity matrix. When this is done correctly, the augmented part of the matrix simultaneously becomes A−1.

As you know, multiplying a matrix by its inverse always gives the Identity matrix. That’s why we can obtain the inverse of A using this method.

 

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