Right, the brute force way is better than sheer guesswork, but it's still too low-level. Kind of like solving the problem in assembly language. The intention of the text is clearly to get one thinking about these operations at a higher level, "at the level of the problem" as we would say in software.

Your WLOG is elegant, though, in that it gives a simpler starting point that's guaranteed to work if anything does. I'll have to remember that trick.

I feel like there is unlikely to be a solution which doesn't involve (1) brute force, or (2) higher-level theory (i.e. eigenvectors and eigenvalues).

Prove me wrong, and I will be tripping over myself to upvote you ;)

Certainly it's artificial to restrict oneself to primitive concepts, unless of course you don't know about eigenvalues yet or forgot it all. :) And it is an exercise from the earliest chapter, so we're supposed to stay in simple-land.

Did you look at my solution (http://news.ycombinator.com/item?id=3129023)? There's no higher-level theory in it. Is it brute force? Not sure. At least it doesn't require solving any element-wise equations.

It's elegant, but as far as I can see, your argument only holds for the single matrix C which you describe.

Oh, absolutely. The exercise is to construct one example.

If you were talking about solving the general case (For any C, find D such that CD = -DC) I can see why you wanted more powerful tools.

oh! that's easier :)