This means the characteristic equation for A is = x^2=1, or x^2-1 = 0...since coefficient of x^1 is 0, so trace of A = 0.
Careful. The equation A^2 - 1 = 0 tells you that the characteristic polynomial of A is one of the following:
a) (x-1)^2 = x^2 - 2x + 1
b) (x+1)^2 = x^2 + 2x + 1
c) (x-1)*(x+1) = x^2 - 1
You need the hypothesis that A is not I or -I to conclude that you are in case (c). [Think this way: If A was the identity matrix, then the trace of A is 2 and not 0, so you certainly need to use the hypothesis somewhere.]
In case someone wants a short refresher on minimal and characteristic polynomials:
The minimal polynomial of a matrix is the least polynomial p(x) so that p(A) = 0. If q(x) is any other polynomial equation with q(A) = 0, then p|q. In this problem we know that A^2 - 1 = 0, so the minimal polynomial of A divides x^2 - 1 = (x-1)(x+1). So the minimal polynomial is either (x-1), (x+1) or (x-1)*(x+1). Now we need a relationship between the minimal and the characteristic polynomials:
- both polynomials are a product of (x - Eigenvalue) factors
- the minimal polynomial divides the characteristic polynomial (since the 0 = characteristic polynomial evaluated at A)
More precisely: the multiplicity of an (x-a) factor in the minimal polynomial is the size of the largest Jordan block of the Eigenvalue a in the Jordan Normal Form, whereas the multiplicity of (x-a) in the characteristic polynomial is sum of the sizes of all Jordan blocks of a.
For example, if the minimal polynomial of an nxn matrix is (x-1), then the characteristic polynomial is (x-1)^n. If the minimal polynomial is (x-1)*(x+1), then the characteristic polynomial is (x-1)^k * (x+1)^(n-k).
It takes some care to go back and forth between minimal and characteristic polynomials. Two matrices with the same characteristic polynomial may have distinct minimal polynomials, or they may have the same minimal polynomial with distinct characteristic polynomials. For example, the matrices
both have characteristic polynomial (x-1)^2, but the minimal polynomial of A is (x-1) while the minimal polynomial of B is (x-1)^2.
On the other hand, the diagonal matrices
have the same minimal polynomial (x-1)*(x-2), but different characteristic polynomials (x-1)^2 * (x-2) and (x-1) * (x-2)^2.