LAUSR

Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space $$\mathbb {R}^{p',q'}$$Rp′,q′, $$n'=p'+q'=2m$$n′=p′+q′=2m, we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over $$\mathbb {R}^{p,q}$$Rp,q, $$n=p+q=p'+q'+1=2m+1$$n=p+q=p′+q′+1=2m+1. Explicit examples are provided for dimensions $$n'=2,4,6$$n′=2,4,6, and the resulting inverses for $$n=n'+1=3,5,7$$n=n′+1=3,5,7. The general result for $$n=7$$n=7 appears to be the first ever reported closed algebraic expression for a multivector inverse in Clifford algebras Cl(p, q), $$n=p+q=7$$n=p+q=7, only involving a single addition of multivector products in forming the determinant.