The correct additional requirement for this problem's statement to be true is that the eigenvalues of A never equal 1.
   
David Jordan

Suppose A is an n x n matrix such that A³ = I and

       We received four different correct solutions from Wengi Shao, Josiah Thornton, Travis Willse, and Nathan Collins for Write As Is. We also received three proofs for the incorrectness of Prove It. One was from Nathan Collins with help from David Jordan and Zack Almquist. The other two were from David Jordan and Travis Willse. The winners of the Dick Koch drawing for the pizza are David Jordan, Nathan Collins, Zach Almquist, and Travis Willse . Congratulations!

         Suppose that A is an nxn matrix with the property that A³ = I, the identity matrix. Prove that the trace of A + A² + I is zero.

The statement of this problem is incorrect. Several of our students caught this. Here are their contributions, edited by Dick Koch.

Nathan Collins, David Jordan & Zack Almquist

        First we need to require that A ¹ I. The problem doesn't specify this, but it is certainly necessary. Next consider this infinite class of matrices

Then tr(A + A² + I) = 0.

Pf.    Treat A as a matrix over C (it won't change the trace). Now, WLOG, assume A is in Jordan Normal Form (it suffices if it is in triangular form). Since A³ = I, each diagonal l_i satisfies  l_i³ = 1. Since l_i ¹ 0,  l² +  l + 1 = 0.

Travis Willse
    The assertion is false. A simple counterexample: A = I. Then, A + A²+ I = 3I, and Tr (A + A² + I) = 3n.
Slightly more interesting counterexamples:

      M_n³ = I_n+2  and  M_n ¹ I_n+2  and tr M_n + tr M_n² + tr I_n+2 = 3n ¹ 0 for n = 1. So there are problems with the problem, and we cannot fix them by merely requiring that A ¹ I.   If complex numbers are allowed, then the following 2 x 2 matrix is a counter example since tr A² + tr A = tr I = 3.

        The above matrices give 1, 1, and 6, respectively,
for Tr (A + A² + I).