Consider the lower triangular matrix A:
Divide the COLUMNS with the diagonal elements in matrix A:
Which gives us matrix B:
Now replace the all the ones on the main diagonal with zeros:
Then calculate matrix powers as follows: Which is exactly as the binomial series: except that here it is applied to a triangular matrix and the result is a new triangular matrix D. Notice that is the identity matrix.
And then finally divide the ROWS in matrix D with the diagonal elements in A: