Math21b, Spring 2009, Linear Algebra and Differential Equations,

Problem: Show that all Pascal matrices like

                1    1    1    1    1

                1    2    3    4    5

                1    3    6    10   15

                1    4    10   20   35

                1    5    15   35   70

have determinant 1. You get the pascal triangle, if you rotate the matrix by 45 degrees. You can generate the matrix as follows:

A[n_]:=Table[Binomial[k+l-2,l-1],{k,n},{l,n}];

Here is an other type of matrices which could be called Pascal triangle matrices. Also they have determinant 1, but because of much simpler reasons.

                1    0    0    0    0

                1    1    0    0    0

                1    2    1    0    0 

                1    3    3    1    0 

                1    4    6    4    1

A cool way to generate it is to write it as the matrix exponential exp(B) = 1+B+B^2/2! + ... of a simpler matrix.

A[n_]:=Table[If[i-j==1,i-1,0],{i,n},{j,n}]; MatrixExp[A[5]]

Please send questions and comments to math21b@fas.harvard.edu