Question

If A=
1 2
0
1
then An =​

Answer 1

Answer:

The product of lower triangular matrices is again lower triangular and on the diagonal they have the products of the diagonal elements.

Thus each power will have 1 on the diagonal and 0 at the top right corner. It suffices to determine the bottom left corner. Set

An=[1xn01]

with x1=1 . We want to exploit the recursive definition of power, so

[1xn+101]=An+1=AnA=[1xn01][1101]=[1xn+101]

and therefore xn+1=xn+1 . This clearly implies xn=n , so finally

An+1=[1n01]

Now try with

F=[0111]

It will be fun.