determinantteja

Determinants are scalars associated with square matrices. In linear algebra, the determinant of an n-by-n matrix A, denoted det(A) or |A|, measures how the linear transformation x -> Ax changes volume and orientation. A determinant is zero exactly when A is singular (its columns are linearly dependent), and nonzero determinants indicate invertibility. In some languages, the plural form used for multiple determinants is determinantteja.

Determinants have several key properties. They are multilinear in the rows (or columns) and change sign with each row swap; det(I) = 1. For any square matrices A and B, det(AB) = det(A) det(B); det(A^T) = det(A); if A is invertible, det(A^{-1}) = 1/det(A). For block-diagonal matrices, det is the product of the determinants of the blocks; for triangular matrices, det equals the product of the diagonal entries.

Computing determinants can be done by the 2×2 formula ad − bc, by cofactor expansion, or by row reduction to upper triangular form while tracking row swaps. In practice, row-reduction methods are common for larger matrices.

Applications of determinants are broad. They determine invertibility, enable Cramer's rule for solving linear systems, and appear in change-of-variables formulas in calculus. They also quantify volumes and oriented volumes, and they contribute to the construction of characteristic polynomials det(A − λI), whose roots are the eigenvalues of A.

Determinants are defined for square matrices over any field (real, complex, etc.). Their magnitude provides volume scaling under the associated linear map, while the sign indicates orientation. A zero determinant signifies a singular matrix.