Determinants scale volume and decide invertibility
One number built from a square matrix tells you whether the matrix is invertible, how a linear map stretches volumes, and—via Cramer's rule—how to solve linear systems. Similar matrices share that number, so it belongs to the transformation itself.
Written det(A), det A, or |A|, the determinant is a scalar function of the entries of a square matrix. One clean characterization: it is the unique map sending matrix products to products of scalars and reading a triangular matrix as the product of its diagonal entries. Hence det(AB) = det(A)det(B). A matrix over a field is invertible exactly when the determinant is nonzero; the adjugate supplies an explicit inverse formula. Similarity preserves determinants, so a linear transformation has a well-defined determinant. For real maps that value is the oriented volume scale factor—positive if orientation stays, negative if it flips, zero if the map collapses dimension.
The determinant is multilinear in rows and columns. Laplace expansion writes it as a combination of smaller cofactor determinants, handy when a row or column is sparse. Practical computation uses Gaussian elimination or singular value decomposition; the Leibniz formula and full Laplace expansion are mostly theoretical. Beyond linear algebra, determinants appear in characteristic polynomials whose roots are eigenvalues, in geometric areas and volumes, in Jacobian change-of-variable formulas, and in the Hessian test for inflection points on plane algebraic curves.
Conventions stay coherent down to tiny sizes: the empty 0×0 matrix has determinant 1, matching empty-product logic, and a 1×1 matrix's determinant is its single entry—facts that keep the adjugate inverse formula honest. For 2×2 matrices the absolute value of the determinant is the area of the parallelogram spanned by the column (or row) vectors, with the sign recording orientation. From empty matrices to Jacobians, the same scalar keeps watching volume and invertibility.
Source: Determinant
