Forum for the GRE subject test in mathematics.

kaiserguy
 Posts: 11
 Joined: Wed Oct 01, 2008 9:16 pm
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by 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. A2I and B2I 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
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by CoCoA » Tue Oct 14, 2008 8:45 pm
I just go to the definitions. B=Q^{1}AQ, so using distributive,
Q^{1}(A2I)Q = (Q^{1}A2Q^{1})Q = Q^{1}AQ  2I = B2I
similar for part III (no pun intended!)

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