Here are two problems that were given to my freshman linear algebra class as optional hard problems:
Problem 1 (given after a discussion of determinants in week 3/4 of the course):
Consider a 9x9 matrix A. We say that A is a Sudoku matrix if it's the valid solution to a Sudoku puzzle. That is if,
1) Every row and every column is a permutation of {1,2,3,4,5,6,7,8,9}.
2) If we write it in block form:
A = \left[\begin{array}{c|c|c} A_1 & A_2 & A_3 \\ \hline A_4 & A_5 & A_6 \\ \hline A_7 & A_8 & A_9 \end{array} \right]
where A_i is a 3x3 matrix, then every A_i has elements {1,2,3,4,5,6,7,8,9}.
Now the problem is:
a) Find a Sudoku matrix with determinant 0.
b) Does there exist a Sudoku matrix with determinant 1. If not then determine the least positive number that a Sudoku matrix can have as a determinant.
Problem 2 (Given after discussing vector spaces, subspaces, linear independence, etc. in week 5/6. This was done as a sort of a contest where the professor picked out the best solutions which is possible due to discussion required in some of the problems.):
The zeros of a real polynomial P(x,y) in two variables is called an algebraic curve. Let p(t), q(t) be real polynomials, and define a real function f : \mathbb{R} \to \mathbb{R}^2 by f(t) = (p(t),q(t)).
a) Prove that the image of f, i.e. the set \{f(t) | t \in \mathbb{R}\}, is contained in an algebraic curve (different from the trivial \mathbb{R}^2 given by P(x,y)=0; in all subsequent questions we shall assume that P(x,y)=0 isn't a valid solution).
b) Let p(t) = t^2, q(t) = t^3 and find a real polynomial P(x,y) in two variables such that the image of f is contained in the algebraic curve determined by P. Do the same question for (p(t),q(t)) = (t^2+t,t^3) and (p(t),q(t)) = (t^2+t,t^2).
c) Consider how to define the degree of a polynomial in 2 variables [at this point we hadn't been introduced to the definition so this is sort of an essay question]. Let \deg(p(t)) = d_p and \deg(q(t)) = d_q. Let P_{min}(x,y) be a polynomial of minimal degree among those defining an algebraic curve that contains the image of f. What can you deduce about the degree of P_{min}?