GRE 0568

Forum for the GRE subject test in mathematics.
kaiserguy
Posts: 11
Joined: Wed Oct 01, 2008 9:16 pm

GRE 0568

Postby kaiserguy » Tue Oct 14, 2008 8:33 pm

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

CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm

Postby CoCoA » Tue Oct 14, 2008 8:45 pm

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!)

kaiserguy
Posts: 11
Joined: Wed Oct 01, 2008 9:16 pm

Postby kaiserguy » Tue Oct 14, 2008 8:49 pm

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.

Nameless
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm

Postby Nameless » Tue Oct 14, 2008 9:57 pm

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)




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