If I+A+⋯+An−1=O, A a square integer matrix, n odd, for what k does det(∑n−1i=kAi)=±1?

manpreet Best Answer 2 years ago

This question is, in a sense, homework. I'm taking a problem-solving seminar which uses questions like these, the first question on the 2010 Virginia Tech Regional Math Competition, as fodder. The syllabus tells me that correct solutions do not factor into grading, so asking this on MSE is kosher.

The exam asks about a particular case:

Let dd be a positive integer and let AA be a d×dd×d matrix with integer entries. Suppose I+A+A2+⋯+A100=OI+A+A2+⋯+A100=O (where I denotes the identity d×dd×d matrix, so II has 1’s on the main diagonal, and OO denotes the zero matrix, which has all entries 00). Determine the positive integers n≤100n≤100 for which An+An+1+⋯+A100An+An+1+⋯+A100 has determinant ±1±1.

I'm really quite stumped on this one. Of course (multiplying by I−AI−A) we will have An=IAn=I. Let me note explicitly that such