  Modern advances in general-purpose computer algebra systems offer solutions
to a variety of problems, which in the past required substantial time
investments by trained mathematicians. An excellent example of such development
are the Clifford algebras. The main objective of the paper is to demonstrate an
utterly algorithmic construction of a Clifford algebra matrix algebra
representation of a non-degenerate signature (p, q). While this is not the most
economical way of implementation, it offers a transparent mechanism of
translation between a Clifford algebra and its faithful real-valued matrix
representation and can be used for automated proof checking. This
representation is used to derive an algorithm for the computation of an
arbitrary multivector inverse as a proof certificate. The proposed algorithm is
a mapping of the Faddeev--LeVerrier--Souriau algorithm for computation of the
characteristic polynomial of matrices.
