  In this paper we look for closed expressions to calculate the number of
colourings of prime knots for given linear Alexander quandles. For this purpose
the colouring matrices are simplified to a triangular form, when possible. The
operations used to perform this triangularization preserve the property that
the entries in each row add up to zero, thereby simplifying the solution of the
equations giving the number of colourings. When the colouring matrices (of
prime knots up to ten crossings) can be triangularized, closed expressions
giving the number of colourings can be obtained in a straightforward way. We
use these results to show that there are colouring matrices that cannot be
triangularized. In the case of knots with triangularizable colouring matrices
we present a way to find linear Alexander quandles that distinguish by
colourings knots with different Alexander polynomials. The colourings of knots
with the same Alexander polynomial are also studied as regards when they can
and cannot be distinguished by colourings.
