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The 54 pages that link to Determinant, each with the reason it gives.

• AreaRelated: The absolute determinant gives the area-scaling factor of a planar linear map.

• Linear algebraRelated: A nonzero determinant identifies an invertible square matrix and a unique solution to its system.

• Coordinate geometryRelated: Determinants test collinearity and calculate areas from coordinate data.

• Cross productRelated: A determinant computes the cross product's components from the input vectors.

• Linear independenceRelated: For a square matrix, a nonzero determinant is equivalent to independence of its columns.

• Rotation matrixRelated: A determinant of 1 distinguishes rotations from reflections among orthogonal transformations.

• Group homomorphismRelated: On invertible matrices, the determinant is a homomorphism to the multiplicative group of nonzero scalars.

• Characteristic polynomialRelated: The characteristic polynomial is defined by taking a determinant.

• Affine geometryRelated: The linear part's determinant detects area scaling and orientation reversal.

• CollinearityRelated: A zero determinant tests whether coordinate vectors span a line rather than a plane.

• Implicit function theoremRelated: A nonzero determinant expresses the theorem’s condition for solving locally for selected variables.

• System of linear equationsRelated: For a square system, a nonzero coefficient determinant guarantees a unique solution.

• Invertible matrixRelated: A square matrix is invertible exactly when its determinant is nonzero.

• Triangle areaRelated: A 2-by-2 determinant of edge vectors gives twice the signed triangle area.

• Cross-ratioRelated: In homogeneous coordinates, brackets of pairs of point vectors express the cross-ratio through determinants.

• LU decompositionRelated: The determinant follows from the diagonal entries of U and the row-swap sign.

• Orthogonal matrixRelated: An orthogonal matrix has determinant either 1 or −1.

• Gram matrixRelated: Principal determinants of Gram matrices encode squared volumes spanned by vector subsets.

• Square matrixRelated: Only square matrices have determinants, which test whether their transformations are invertible.

• Eigenvalue problemRelated: A matrix has an eigenvalue when its shifted matrix has zero determinant.

• Cayley–Hamilton theoremRelated: Nineteenth-century determinant methods supplied the characteristic polynomial central to the theorem.

• Exterior algebraRelated: The coefficient of a top-degree wedge product is a determinant.

• Matrix inverseRelated: A matrix has an inverse exactly when its determinant is nonzero.

• Permutation matrixRelated: A permutation matrix’s determinant is the sign of its encoded permutation.

• Diagonal matrixRelated: For a diagonal matrix, the determinant is the product of its diagonal entries.

• James Joseph SylvesterRelated: Determinants were central to Sylvester’s matrix work and his study of algebraic forms.

• Matrix theoryRelated: It detects singularity and connects algebraic structure to geometric volume.

• Minkowski's theorem (geometry of numbers)Related: A basis matrix's determinant gives the lattice covolume used in the theorem.

• Triangular matrixRelated: A triangular matrix’s determinant is the product of its diagonal entries.

• CoplanarityRelated: A zero determinant in a coordinate test signals that the relevant vectors do not span three-dimensional volume.

• Eigenvalues and eigenvectorsRelated: The condition det(A − λI) = 0 makes the eigenvector equation have nonzero solutions.

• Line–line intersectionRelated: A nonzero determinant of direction or coefficient data signals a unique planar intersection.

• Pappus's hexagon theoremRelated: A zero determinant expresses the collinearity of the three intersection points.

• Gershgorin circle theoremRelated: Its characteristic-polynomial equation defines the eigenvalues bounded by the disks.

• Cartan–Dieudonné theoremRelated: Each reflection has determinant −1, so factor parity determines an isometry’s determinant.

• De Gua's theoremRelated: Determinants and Gram matrices can calculate areas from edge vectors.

• Dittert conjectureRelated: It connects vertex coordinates to simplex volume and explains affine scaling.

• FKT algorithmRelated: The Pfaffian can be computed through a determinant, enabling efficient numerical evaluation.

• Monsky's theoremRelated: Triangle area is half the absolute determinant of two edge vectors.

• Unit (ring theory)Related: A matrix over a commutative ring is invertible exactly when its determinant is a unit.