determinantrelated

Determinantrelated refers to concepts, results, and methods that are inherently tied to the determinant of a square matrix. The determinant is a scalar associated with an n-by-n matrix that describes how the corresponding linear transformation scales volumes, and it detects invertibility and orientation. It is zero exactly when the transformation is singular.

The determinant can be defined by a sum over permutations, det(A) = sum_{σ∈S_n} sgn(σ) ∏_{i=1}^n a_{i,σ(i)}; it can also be computed by row reduction, LU decomposition, or cofactor expansion. For a 2×2 matrix [ [a,b], [c,d] ], det = ad − bc.

Key properties include multiplicativity det(AB) = det(A) det(B), and det(A^T) = det(A). Elementary row operations affect the determinant in predictable ways: swapping two rows changes the sign, multiplying a row by a scalar scales the determinant by that scalar, and adding a multiple of one row to another leaves it unchanged. Similar matrices have the same determinant: det(SAS^{-1}) = det(A). The determinant is a polynomial in the entries of A of total degree n, and it equals the product of eigenvalues: det(A) = ∏ λ_i.

The determinant detects singularity: det(A) = 0 iff A is noninvertible. If det(A) ≠ 0, A has an inverse given by A^{-1} = adj(A)/det(A). In calculus, the Jacobian determinant appears as the factor that converts volumes under a change of variables.

Applications include solving linear systems (Cramer’s rule), volume and orientation computations, and change of variables in integration, as well as roles in physics and computer graphics.