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.