M  A  T  H 
2  1  B 
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 70have 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+l2,l1],{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 1A 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[ij==1,i1,0],{i,n},{j,n}]; MatrixExp[A[5]]
Please send questions and comments to math21b@fas.harvard.edu
Math21b (Exam Group 1) Oliver Knill  Spring 2009 
Department of Mathematics 
Faculty of Art and Sciences 
Harvard University
