Determinant

A scalar-valued function det: M_n(F) → F on the n×n square matrices over a field or commutative ring F characterized by multilinearity in rows (or columns), alternation (sign change under row swap) and normalization det(I)=1; it measures oriented volume scaling and decides invertibility.

A scalar-valued function assigned to a square linear operator or matrix that records oriented volume scaling and indicates invertibility; algebraically it is an alternating multilinear form of the columns (or rows).

A scalar-valued polynomial invariant assigned to a square matrix or endomorphism that encodes whether the linear map is invertible and how oriented volumes are scaled by the map.

A scalar invariant associated to a square matrix or endomorphism that encodes volume scaling factor, orientation sign, and invertibility: a matrix is invertible iff its determinant is nonzero.