Determinant

The determinant assigns a single number to every square matrix and measures the signed factor by which its linear transformation changes volume.¹ For a 2 × 2 matrix, A = [[a,b],[c,d]], the determinant is det(A) = ad − bc. In two dimensions its absolute value is the area-scaling factor, while its sign records whether orientation is preserved or reversed. In three dimensions, the absolute value gives the volume scale of the parallelepiped formed by the transformed basis vectors.

For an n × n matrix, the determinant is defined by the Leibniz formula as a signed sum over all permutations of the columns. Although this formula is conceptually central, direct evaluation becomes inefficient as n grows; elimination-based methods are normally preferred. The determinant belongs to the field containing the matrix entries and changes predictably under elementary row operations.

The determinant is multilinear in the rows or columns and vanishes whenever two rows or columns are equal or linearly dependent.² These properties, together with normalization det(I) = 1, characterize it uniquely as an alternating multilinear function. Swapping two rows changes the sign; multiplying one row by a scalar multiplies the determinant by that scalar; and adding a multiple of one row to another leaves it unchanged.

For compatible square matrices, det(AB) = det(A)det(B), a relation that connects composition of transformations with multiplication of volume factors. If Gaussian elimination converts A to triangular form, the determinant is the product of the diagonal entries, adjusted for row swaps and row scalings. Cofactor expansion is useful for small matrices and for theoretical identities, but it is generally poor for large numerical problems.

A square matrix is invertible exactly when its determinant is nonzero.¹ Equivalently, its columns form a basis, its nullspace contains only the zero vector, and the linear system Ax = b has a unique solution for every b. The equation det(A − λI) = 0 defines the characteristic polynomial; its roots are the eigenvalues, counted with algebraic multiplicity, and their product equals det(A).

Determinants therefore appear in the characteristic equation, the change-of-variables formula in multivariable calculus, and Cramer's rule. In physics and engineering, Jacobian determinants describe local coordinate changes, while Gram determinants give squared volumes and test linear independence. Cramer's rule is mathematically elegant but usually less efficient and less numerically stable than factorization methods for solving large systems.

Determinants also arise in combinatorics, differential equations, probability, and geometry, often through specialized variants rather than ordinary matrix formulas. The Gram determinant of vectors is the square of the volume of their parallelepiped, so it is nonnegative and vanishes precisely when the vectors are linearly dependent.³ The Wronskian, a determinant built from functions and their derivatives, can help test linear independence of solutions to differential equations, though a zero Wronskian requires careful interpretation.

In graph theory, determinants of matrices associated with a graph count spanning trees through the matrix-tree theorem. In numerical linear algebra, the sign and logarithm of a determinant are often computed separately, because determinants can overflow or underflow even when the underlying matrix is well behaved. For complex matrices, the determinant may be complex; for a unitary matrix its absolute value is one, a fact linked to volume preservation in complex coordinates.

Square matrix

A matrix with the same number of rows and columns; ordinary determinants are defined for square matrices.

Minor

The determinant obtained after deleting a specified row and column from a matrix.

Cofactor

A signed minor used in Laplace expansion and inverse-matrix formulas.

Jacobian determinant

The determinant of a derivative matrix describing local scaling under a coordinate transformation.

Gram determinant

The determinant of a Gram matrix; it equals the squared volume of the associated parallelepiped.