  The Brioschi formula expresses the Gaussian curvature $K$ in terms of the
functions $E, F$ and $G$ in local coordinates of a surface $S$. This implies
the Gauss' theorema egregium, which says that the Gaussian curvature just
depends on angles, distances, and their rates of change.
  In most of the textbooks, the Gauss' theorema egregium was proved as a
corollary to the derivation of the Gauss equations, a set of equations
expressing $EK, FK$ and $GK$ in terms of the Christoffel symbols. The
Christoffel symbols can be expressed in terms of $E$, $F$ and $G$. In
principle, one can derive the Brioschi formula from the Gauss equations after
some tedious calculations.
  In this note, we give a direct elementary proof of the Brioschi formula
without using Christoffel symbols. The key to the proof are properties of
matrices and determinants.
