determinates

Determinants, also historically referred to as determinates, are scalars associated with square matrices. For an n×n matrix A over a field, the determinant det(A) (often written |A|) encodes the scaling factor of the linear transformation x → Ax. Geometrically, det(A) equals the signed n-dimensional volume of the parallelepiped spanned by the columns of A; det(A) = 0 precisely when the columns (or rows) are linearly dependent, in which case A is singular.

For a 2×2 matrix [a b; c d], det(A) = ad − bc. In general, the determinant is multilinear in the rows (or columns) and is alternating: swapping two rows changes its sign.

Key properties include: det(AB) = det(A) det(B); det(A^T) = det(A); det(kA) = k^n det(A) for a scalar k and size n; det(A^{-1}) = 1/det(A) when A is invertible. These properties underpin many computational techniques and theoretical results.

Computation methods vary. A direct Laplace expansion along a row or column is conceptually straightforward but grows expensive for large n. Sarrus’ rule applies to 3×3 matrices. More commonly, one reduces A to upper triangular form via Gaussian elimination and computes det(A) as the product of the diagonal entries, adjusting for any row swaps (which flip the sign) and any row scalings (which multiply the determinant by the scaling factor).

Applications of determinants are broad. They provide solutions to linear systems (via Cramer’s rule in principle), determine invertibility, and quantify changes of variables in multi-variable integrals. They also give geometric insight into the effect of linear maps on volume and orientation.