I have a question. Prove that the product of an [arbitrary] number of upper triangular matrices of [arbitrary] size with [undetermined] upper triangular entries is upper triangular using induction? Should I use transfinite induction? I don't even know where to start!
How do you even prove the statement: "prove that the product of n upper triangular matrices with undetermined upper triangular entries is upper triangular"?
I know that the product of n upper triangular matrices with all upper triangular entries = 1 is
[1 3 6 ... n(n+1)/2 0 1 3 ... (n-1)n/2 0 0 1 ... (n-2)(n-1)/2 . . . . . .
with the pattern continuing such that a_(nn) = [n-(n-1)][n-(n-2)]/2.