Gaussian Elimination without pivoting

Given a matrix A (nXn), define the a series matrices as following:

A_1 = | a_{1,1} |
        
         | a_{1,1} a_{1,2} |
A_2 = | a_{2,1} a_{2,2} | 

         | a_{1,1} a_{1,2} a_{1,3} |
A_3 = | a_{2,1} a_{2,2} a_{2,3} |
         | a_{3,1} a_{3,2} a_{3,3} |

…

Now, it could be verified that we can using Gaussian Elimination for matrix A without pivoting if and only if A_1, A_2, \cdots, A_n are all non-singular.

I don’t really understand Gaussian Elimination!

That’s right! You think you know it, but unfortunately that might be not true!

The baby-level thing – Gaussian Elimination is quite straightforward.  But the following things might be not that familiar to you. First of all, GE could give us the LU decomposition for most matrices. Second, GE is connected to the so-called completing squares in some interesting ways.

Anyway, I have to stop this post.