  In this paper, we solve the problem of computing the inverse in Clifford
algebras of arbitrary dimension. We present basis-free formulas of different
types (explicit and recursive) for the determinant, other characteristic
polynomial coefficients, adjugate, and inverse in real Clifford algebras (or
geometric algebras) over vector spaces of arbitrary dimension $n$. The formulas
involve only the operations of multiplication, summation, and operations of
conjugation without explicit use of matrix representation. We use methods of
Clifford algebras (including the method of quaternion typification proposed by
the author in previous papers and the method of operations of conjugation of
special type presented in this paper) and generalizations of numerical methods
of matrix theory (the Faddeev-LeVerrier algorithm based on the Cayley-Hamilton
theorem; the method of calculating the characteristic polynomial coefficients
using Bell polynomials) to the case of Clifford algebras in this paper. We
present the construction of operations of conjugation of special type and study
relations between these operations and the projection operations onto fixed
subspaces of Clifford algebras. We use this construction in the analytical
proof of formulas for the determinant, other characteristic polynomial
coefficients, adjugate, and inverse in Clifford algebras. The basis-free
formulas for the inverse give us basis-free solutions to linear algebraic
equations, which are widely used in computer science, image and signal
processing, physics, engineering, control theory, etc. The results of this
paper can be used in symbolic computation.
