Pascal Matrices

In the Final exam, we looked at Pascal triangle matrices. So, here is the Pascal triangle, then the Pascal triangle, in which the integers are seen modulo 2 (we look at the remainder when dividing by 2, which means that we put 1 if the number is odd and 0 if the number is even). It was generated with the following line:

M=9; A=Table[ Binomial[n,k],{n,0,M-1},{k,0,M-1}]; Mod[A,2]

1  0  0  0  0  0  0  0  0
1  1  0  0  0  0  0  0  0 
1  2  1  0  0  0  0  0  0 
1  3  3  1  0  0  0  0  0 
1  4  6  4  1  0  0  0  0 
1  5 10 10  5  1  0  0  0  
1  6 15 20 15  6  1  0  0 
1  7 21 35 35 21  7  1  0 
1  8 28 56 70 56 28  8  1

1  0  0  0  0  0  0  0  0
1  1  0  0  0  0  0  0  0
1  0  1  0  0  0  0  0  0
1  1  1  1  0  0  0  0  0
1  0  0  0  1  0  0  0  0
1  1  0  0  1  1  0  0  0
1  0  1  0  1  0  1  0  0
1  1  1  1  1  1  1  1  0
1  0  0  0  0  0  0  0  1

Here is a larger version for n=2⁹ = 512. The picture of the matrix converges to a fractal called the Sierpinski triangle.

In the exam, we have computed the first column of the inverse of the Pascal matrix. Here is a picture of the inverse for n=2048. The inverse takes values 0,1,-1 only.