det, dot, cross, transpose and resultant as functions in an expression #2147

Modenov's Problems in Geometry (#2131) computes with determinants and vector products throughout:

• oriented areas, and collinearity, concurrency and similarity tests;
• the vector and scalar triple products, and the pseudo-scalar product in the plane;
• eliminating a variable from two polynomials (Chapter III, Problem 41).

Each of these exists in C#, as Matrix.Determinant and MathS.Polynomials.Resultant. None can be written in an expression:

• det(M) is read as det * M (conj, arg, re, im, bar and det are read as products, not refused like conjugate #2136);
• there is no cross, dot or resultant in the parser;
• ]T transposes only a bracket literal, so ([1, 2])T is a product with a variable T.

Wanted, on concrete matrices and vectors:

• det(M), dot(a, b), cross(a, b) in three dimensions;
• the plane's pseudo-scalar product and its rotation through π/2, which the book writes (a, b) and [a];
• transpose(M), and resultant(p, q, x).

These are functions over the components, with no new node. #105 (3.0) remains the question of cross and dot on symbols that stand for vectors, which needs #1237's typed variables.

The names and their LaTeX are a choice to make once. For example, dot as a \cdot b and cross as a \times b. The book's (a, b) and [a, b] are not available, because the parser reads [a, b] as a vector and [a; b] as an interval.