M A T H 2 1 B
Mathematics Math21b Spring 2017
Linear Algebra and Differential Equations
Exhibit: Pascal Matrices
Course Head: Oliver Knill
Office: SciCtr 432

# 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.
Please send questions and comments to knill@math.harvard.edu
Math21b Harvard College Course ID:110989| Oliver Knill | Spring 2017 | Department of Mathematics | Faculty of Art and Sciences | Harvard University, [Canvas, for admin], Twitter