I'm looking for challenging problems in elementary linear algebra, i.e. mainly about matrices in the field of real numbers. Can anyone provide some references?
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Pretty much any linear algebra book would be an obvious start...
I like MIT's OpenCourseWare, and they have an excellent course on Linear Algebra. You could certainly find plenty of exercises there, and there is particularly relevant material in the "Related Resources" section. Paul's Online Notes also has a linear algebra section. There's also "A First Course in Linear Algebra", a free online book which has an interactive Sage version available.
A Google search would also reveal plenty of free online books from which to pull examples:
It would probably be best to choose a book targeted toward your eventual goal - linear functional analysis, number theory, an introduction to abstract spaces? From the wording of your question, it sounds like you are just focusing on the mechanics of elementary row operations, determinants and the like over the reals. I have not found that any of these sorts of problems are ever that challenging, the more challenging parts of linear algebra are generally the proofs.
If you are interested in proofs, I would be remiss if I did not mention my favorite book on Linear Algebra, Linear Algebra Done Right. I hesitated to mention it immediately, since it does not focus exclusively on the reals, however it is an excellent source for proof based questions and provides and excellent reference for well written proofs within the text.
Please look up the link I gave you.I think the questions for the 11th grade contain a bit of linear algebra.I myself have learned very little of it, so I cannot be sure if that will be challenging enough for you.Hope that helps.The book Putnam and Beyond has problems in linear algebra as well. http://www.amazon.com/Putnam-Beyond-Razvan-Gelca/dp/0387257659
Please have a look at the answers at this MO link, not all the stuff is elementary, but you will find a good list.