  I present a universal method, called pivotal condensation, for calculating
stoichiometric factors of chemical reactions. It can be done by hand, even for
rather complicated reactions. The main trick, which I call kernel pivotal
condensation (ker pc), to calculate the kernel of a matrix might be of
independent interest. The discussion is elaborated for matrices with entries in
a principal ideal domain $R$. The ker pc calculates a basis with coefficients
in $R$ for the kernel of a matrix, seen as e $Q$-vector space, where $Q$ is the
quotient field of $R$. If $W$ is a free saturated $R$-submodule of $R^n$ I
address the question how to modify an $R$-basis of the $Q$-vector subspace
$Q\otimes _R W$ over the quotient field $Q$ to obtain a basis of the $R$-module
$W$. I also indicate how one can solve inhomogeneous linear systems, invert
matrices and determine the four subspaces using pivotal condensation. I
formulate the balancing by inspection method that is widely used to reduce the
size of a linear system arising in chemical balancing in mathematical language.
