detu

detU denotes the determinant of a square matrix U. For an n×n matrix U, detU is a scalar that captures key properties of the linear transformation represented by U, including invertibility and volume scaling.

Key properties include: det(AB) = detA detB for square matrices A and B; det(A^T) = detA; det(kA) = k^n detA for a scalar k; det(I) = 1. The determinant is zero exactly when the matrix is singular, and it is nonzero precisely when the matrix is invertible. Row and column operations affect the determinant in characteristic ways: adding a multiple of one row to another leaves the determinant unchanged; swapping two rows multiplies det by -1; multiplying a row by a scalar c multiplies det by c. For a triangular matrix, det equals the product of its diagonal entries. If U is unitary (or orthogonal in the real case), then |detU| = 1.

Computation methods include the cofactor expansion, or more practical approaches such as LU decomposition: detU equals the product of the diagonal entries of the L and U factors, up to a sign that accounts for row interchanges. The determinant also equals the product of the eigenvalues of U, counting multiplicities, and provides insight into spectral properties.

Applications of detU appear across linear algebra and analysis, including solving linear systems (e.g., Cramer's rule), change of variables in integrals, and assessing linear independence and volume changes under transformations.

Example: for a 2×2 matrix U = [[a, b], [c, d]], detU = ad − bc.