TITLE: Cat quiz (to solve with the determinant)
QUESTION [18 upvotes]: Consider $100$
  cats and
  $100$
  food bowls containing
  cat food of
  $100$
  different
  brands.
Every cat
  likes an odd amount of brands.
For each two
  cats, there is
  an even amount
  of brands both
  cats like.
Show that one can
  distribute the
  $100$
  food
  bowls
  to the
  $100$
  cats such that every cat is happy.
Hint:
  It's a
  determinant
  exercise
  over
  the field
  $\mathbb{F}_2 = 
\mathbb{Z}/2\mathbb{Z}=
\{
[0],[1]
\}$.

How can this be done using
the determinant? Thanks in advance!

REPLY [18 votes]: Let $M$ be the $100 \times 100$ matrix over $\mathbb{F}_2$ that has, at index $(i, j)$, a $1$ if cat $i$ likes food brand $j$ and a $0$ otherwise. Now, I like to see the Leibniz expansion of the determinant as a sum of products over certain "paths" through this matrix. Really, the problem is to show that at least one of these paths has all ones---that is, has non-zero product. This is certainly the case if the determinant of the whole matrix is non-zero. Can you see why this must be true? 
One approach: Apply Gaussian elimination. Note that elementary operations on the rows of $M$ preserve the property that every row has an odd number of ones and every pair of rows share an even number of ones, but a matrix in reduced echelon form with this property can't be singular.
A better approach suggested by @omnomnomnom: check that $M M^T = I$.