determinant

The determinant is a function that assigns to every square matrix A a scalar det(A). For an n-by-n matrix over a field, the determinant captures both algebraic and geometric information about the linear transformation represented by A. It is the signed volume scaling factor that the transformation applies to the unit n-cube.

Key properties include det(AB) = det(A) det(B) and det(A^T) = det(A). Row and column operations affect determinants in predictable ways: adding a multiple of one row to another leaves det unchanged; swapping two rows or columns multiplies det by -1; scaling a row by a nonzero factor k scales det by k. For triangular matrices, the determinant is the product of the diagonal entries.

Determinants provide fundamental criteria and interpretations. det(A) ≠ 0 if and only if A is invertible; in that case Ax = b has a unique solution for every b. The determinant is also the product of the eigenvalues of A, counting algebraic multiplicities, and it plays a central role in formulas such as Cramer’s rule for solving linear systems.

Computationally, determinants are often found by Gaussian elimination to triangular form, keeping track of row swaps and scaling factors. The determinant of a triangular matrix equals the product of its diagonal entries, which makes the triangular form a convenient route to the determinant.