Q. Suppose A and B are nxn invertible matrices, where n>1 and I is the identity nxn matrix. If A and B are similar matrices, which of the following are true?

I. A-2I and B-2I are similar matrices

II. A and B have the same trace

III. A^-1 and B^-1 are similar matrices

(A)I only (B) II only (C) III only (D) I & III only (E) I,II,III

The answer is E. I get that II is true. I know similar matrices share alot of properties but can someone prove both I and III.

Again, thanks in advance,

David