In algebra, the determinant is a special number associated to any square matrix, that is to say, a rectangular array of numbers where the (finite) number of rows and columns are equal. The fundamental geometric meaning of a determinant is a scale factor for measure when the matrix is regarded as a linear transformation. Thus a 2 × 2 matrix with determinant 2 when applied to a set of points with finite area will transform those points into a set with twice the area. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra. A matrix is invertible if and only if its determinant is non-zero.
The determinant of a matrix A, is denoted det(A) or |A|. It is also common to denote the determinant with elongated vertical bars:
- The determinant of a matrix is
For a fixed nonnegative integer n, there is a unique determinant function for the n×n matrices over any commutative ring R. In particular, this unique function exists when R is the field of real or complex numbers.
The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.The 2×2 matrix
has determinant
- det A = ad − bc.
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