I am collecting interesting maths exercises that can be solved by undergraduate students. One can find a lot of them in textbooks, on the internet, in old worksheets of one's own student time... and sometimes one is lucky and invents a new exercise. Here is a linear algebra question that came up in my mind last week end. I like it because the solution I found needs no deep theorem but only some basic understanding of linear algebra:

What is the biggest integer k, such that every affine sub-space of codimension k in the space of n x n matrices contains an invertible matrix?

Recall: The codimension of a sub-space is the difference between the dimension of the ambient space and the one of the sub-space. In other words, it is the number of equations necessary to describe the subspace (every equation taking away one degree of liberty). For example, in our common three-dimenssional space the codimension of a line is 2 and the codimension of a plane is 1.