Explicit Formula for Inverse and Determinant in Geometric Algebras over Odd-dimensional Vector Spaces

In this paper, we present explicit formulas for the inverse and determinant in geometric (Clifford) algebras over vector spaces of dimension $n=7$. The derivation of these formulas is made possible by generalizing the concept of conjugation to basis conjugation operations. We further develop a general method for constructing such formulas over odd-dimensional spaces from the known even-dimensional case. To validate computational utility of the results, we provide a numerical implementation of the formulas. The code implementation is available at the repository github.com/kamranuz/clifford_7d. These formulas extend previous results for lower dimensions and offer new insights for applications in mathematical physics and computational geometry.