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Determinants are a fundamental concept in linear algebra, representing the scalar value that quantifies the geometric properties of a linear transformation. They are used to determine various characteristics of matrices, such as invertibility, rank, and orientation. The determinant of a matrix is denoted by det(A) or |A| for a square matrix A. For a 2x2 matrix, the determinant is calculated as follows:

det(A) = ad - bc

where A is the matrix:

A = [a b]

[c d]

For larger matrices, the determinant can be calculated using various methods, including expansion by minors, cofactor expansion, and the use of LU decomposition. The determinant of a matrix is zero if and only if the matrix is singular (non-invertible). Additionally, the determinant of a product of matrices is equal to the product of their determinants:

det(AB) = det(A) * det(B)

Determinants also play a crucial role in the study of eigenvalues and eigenvectors, as well as in the solution of systems of linear equations. In geometry, determinants are used to calculate the area of parallelograms and the volume of parallelepipeds. The concept of determinants extends to more abstract settings, such as vector spaces and multilinear algebra.