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We construct an extension of Gaussian elimination to show that if$\mathbb{F}$ is a topological field, then there is a transitive, free, andcontinuous action of a natural quotient of $GL_k(\mathbb{F}) \timesGL_{k+1}(\mathbb{F})$ on the set $M_k(\mathbb{F})$ of $2 \times k \times (k+1)$hypermatrices over $\mathbb{F}$ with nonzero hyperdeterminant.  We use this action to answer a number of questions including determining thehomotopy groups of $M_k(\mathbb{C})$, counting elements of $M_k(\mathbb{F}_q)$(generalizing an unpublished result of Lewis and Sam), and computinghyperdeterminants for $2 \times k \times (k+1)$ hypermatrices in $O(k^4)$ time,which we use to compute explicit formulas in some special cases.

Nondegenerate $2 \times k \times (k+1)$ Hypermatrices