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GRE 0568

Posted: Tue Oct 14, 2008 8:33 pm
by kaiserguy
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

Posted: Tue Oct 14, 2008 8:45 pm
by CoCoA
I just go to the definitions. B=Q^{-1}AQ, so using distributive,
Q^{-1}(A-2I)Q = (Q^{-1}A-2Q^{-1})Q = Q^{-1}AQ - 2I = B-2I

similar for part III (no pun intended!)

Posted: Tue Oct 14, 2008 8:49 pm
by kaiserguy
Cool. Thanks for your help on the inverse function one as well. You're starting to make me feel quite stupid. I suppose questions are always easier when you know the answer.

Posted: Tue Oct 14, 2008 9:57 pm
by Nameless
For II,
use the fact that tr(AB)=tr(BA) for all matrices A,B
since A=QBP where P=Q^(-1)
then tr (A)=tr(QBP)=tr(BPQ)=tr(B)