Um, you can't. Information tossed into the bit bucket is irretrievably lost.
In your example, A1 is upper triangular, with 0 on the main diagonal. Powers of such a matrix will go to an identically zero matrix.
A1
A1 =
0 1 1 1 1 1
0 0 1 1 1 1
0 0 0 1 1 1
0 0 0 0 1 1
0 0 0 0 0 1
0 0 0 0 0 0
>> A1^2
ans =
0 0 1 2 3 4
0 0 0 1 2 3
0 0 0 0 1 2
0 0 0 0 0 1
0 0 0 0 0 0
0 0 0 0 0 0
>> A1^3
ans =
0 0 0 1 3 6
0 0 0 0 1 3
0 0 0 0 0 1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
>> A1^6
ans =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
As you can see, each successive power kills off one more diagonal. After the 6th power, it is all gone. That last is not only all zero, it is IDENTICALLY vzero.
There is no scheme in the universe that will then allow you to take a fully zero matrix, and then to infer what was some parent matrix that was in fact used to produce it, as there would be infinitely many such solutions. In fact, A1^7==A1^123. All are identically zero. So you cannot also infer what power was used, because any power of 6 or above would do the same on such a matrix.
You might want to do some reading about nilpotent matrices.
You can't get blood from a rock. Well, you can try, but the only blood you will see is that coming from your own knuckles as you pound away.
