The following question is in the exercise of my text book.

A is 4x4 matrix with 1,-1,2 and -2 as its four eigen values. If B = A^4 - 5A^2 + 5I, which of the following is true?

a) det(A+B)= 0 b) trace (A-B) = 0 c) det(B) = 1 d) trace of (A+B) = 4.

I know for sure that trace of A is 0 as the sum of the eigen values is the same.

But i'm not sure how to arrive at the conclusion for A-B without a little more help in defining the matrix B.