Just took the Math GRE
Who knew a multiple choice math test could be so hard? I skipped 19 out of 66 questions. Yet somehow, I feel I did well: maybe it is my skepticism about the capabilities of most math students, combined with the fact the test is ‘rescaled’. Of course, this may be completely unfounded skepticism, since only 2500 students or so take the test, so I may be in competition with the cream of the crop. That makes me all the more eager to see how I fared!
Here are two of the harder problems, that I had no idea how to approach:
- How many invertible 2-by-2 matrices exist over a finite field with q elements?
- If
is an 
-by- matrix such that 
, which of the following are true: 
, 
, or 
?
Papadakis showed me that a brute-force counting technique may be applicable to resolving the first question– a fact I don’t like, but I can appreciate the approach– and constructed an elegant argument to show that, in the second problem, we can make the statements about the trace and determinant, but despite this, is not necessarily the identity!
Possibly relevant posts:
- Some Schatten norm stuff (6/17/2008)
- Math skills vs. programming skills (1/25/2005)
- Hilbert space class (7/7/2005)
Just took the Math GREs as well. Did you get the one with the non-abelian group, and they ask which is possible: a^2= b^2 aba=u and some other choices, can’t really remember. Also, the analysis questions sucked.
Comment by NB — 11/12/2005 @ 4:12 pm
Aye, I took the test today too. Isn’t the test “rescaled” with the last four years of scores too?
For the number of invertible 2×2 matrices, fortunately I had done this problem before. In general, the order of GL(n,q) can be computed by counting the number of vectors that fit in the first column, then the number of vectors that fit in the second column (without being a scalar multiple of the first) and so on. To double check myself I computed by hand the number of matrices in GL(2,2) and compared with the choices, and only one matched up. In general the equation is:
|GL(n,q)| = (q^n-1)(q^n-q)…(q^n-q^{n-1})
In any case I skipped a good number of questions, and I can easily recall questions that I accidentally marked the wrong answer; for example, the problem with the infinite sum of matrices, I believe I marked the matrix with the sin’s and cos’s, but I think it should have been the one with sinh and cosh.
Comment by Brandon Williams — 11/12/2005 @ 5:56 pm
For the abelian group thing, I pretty much eliminated it down to the first two, but by that point I had spent too much time on it, so I chose the first because it looked more plausible. The analysis questions do suck: I’m pretty sure I’d paid more attention in the grad real analysis course I’m taking, I could have gotten the one with the convergence of the limits and the integrals of those limits (or something like that).
I didn’t even attempt the matrix exponential question– Brandon could you email me it, or post it? I’d like to work on it. Also, thanks for the tip on the general linear group. I didn’t even think to formulate the question in terms of the order of the general linear group… not that it would have helped me then, but I noticed the formula you’re talking about on Planet Math just now. I’ve never seen that before! But, then again, I was barely introduced to the GL/SL , much less useful theorems regarding them.
Despite it all, I enjoyed the test.
Comment by Alex — 11/14/2005 @ 1:28 pm
Here is the matrix problem, if I remember correctly:
Let A = {{0,1},{1,0}}, and define exp tA by
A + sum_{n=1}^\infty t^n/n! A^n
The choices were
{{sin t,cos t},{cos t,sin t}}
{{sinh t,cosh t},{cosh t,sinh t}}
{{e^t, e^{-t}, {e^{-t},e^t}}
and some other combinations of matrix entries of t and e^t.
Comment by Brandon Williams — 11/15/2005 @ 8:46 am
Hey, I am going to be taking the math gre soon and just wanted to know what are some good study materials. I have the princeton review, and the ets mathematics test review thats published by REA and the online stuff. Is there anything else?
Also if you have any personal idea of what to focus on that would be cool.
Thanx
Comment by Aori Nevo — 8/8/2006 @ 10:32 pm