MathDB

Let

A

be a

2 \times 2

matrix with integer entries and

\det A \neq 0

. If the sequence

\operatorname{tr}(A^n)

, for

n = 1, 2, 3, \ldots

, is bounded, show that A^{12} = I \text{or} (A^2 - I)^2 = O. Here,

I

and

O

denote the identity and zero matrices, respectively, and

\operatorname{tr}

denotes the trace of the matrix (the sum of the elements on the main diagonal).

Problem Statement