DetP

detP typically denotes the determinant of a square matrix P. In linear algebra, the determinant is a scalar that encodes certain properties of the linear transformation represented by P, including volume scaling, orientation, and invertibility.

The determinant can be defined in several equivalent ways, such as through a permutation expansion, via cofactors and minors, or computed efficiently by LU decomposition. For a 2x2 matrix P = [[a, b], [c, d]], det(P) = ad − bc. For larger matrices, formulas exist but are generally computed algorithmically rather than by hand. In numerical practice, determinants are often obtained using decomposition methods that run in cubic time with respect to the matrix size.

Key properties include: det(AB) = det(A) det(B) for square matrices A and B; det(A^T) = det(A); det(kA) = k^n det(A) for an n x n matrix; det(I) = 1; det(A) = 0 if and only if A is singular (not invertible). The determinant equals the product of the matrix’s eigenvalues, counting multiplicities, and its sign indicates whether a linear map preserves or reverses orientation.

Interpretation and uses: the absolute value |det(P)| gives the volume scaling factor of the linear transformation associated with P. A nonzero determinant confirms invertibility. Determinants are used in solving linear systems (Cramer’s rule), change of variables in integration, and in characterizing matrix properties in algebra and geometry. In many contexts, detP is simply shorthand for det(P), with the exact meaning depending on the surrounding domain and conventions.

Example: for P = [[2, 1], [3, 4]], det(P) = 2×4 − 1×3 = 5.