tripledetermines

Tripledeterminates is not a standard, widely adopted term in mathematics. As a result, there is no single formal definition that is universally accepted. In informal or niche discussions, the phrase can be used to refer to one of several related ideas that extend or parallel determinants to a triple of matrices or to a triadic collection of determinant invariants.

One common informal usage is to speak of a triad of determinants associated with three square matrices A, B, and C, i.e., det(A), det(B), and det(C). In this sense, “tripledeterminates” would simply denote the three scalar invariants together, rather than a new single operator.

A more substantial but still informal interpretation imagines a tri-linear or triadic generalization of determinant concepts. In established linear-algebraic language, a closely related notion is the mixed determinant (also called the polarized determinant in some contexts), which generalizes determinants to a family of n×n matrices by examining the coefficient of a multi-parameter polynomial det(t1 A1 + … + tn An). While this construction is well defined and studied, it is typically referred to by its standard name—mixed determinant—rather than by the term tripledeterminates.

Because the determinant is a homogeneous polynomial of degree n, a universal, single “tripledeterminant” operation on three matrices is not standard across all sizes n. If any such concept is adopted, it is usually under a more precise formulation or within a specific mathematical framework, such as a polarization or a block-matrix construction, rather than under a single fixed name.

See also: determinant, mixed determinant, multilinear forms, polarization, block matrices.