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Determinant 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. The determinant det A can be viewed as the oriented area of the parallelogram with vertices at (0,0), (a,b), (a + c, b + d), and (c,d). The oriented area is the same as the usual area, except that it is negative when the vertices are listed in clockwise order. Further, the parallelogram itself can be viewed as the unit square transformed by the matrix A. The assumption here is that a linear transformation is applied to row vectors as the vector-matrix product xTAT, where x is a column vector. The parallelogram in the figure is obtained by multiplying matrix A (which stores the co-ordinates of our parallelogram) with each of the row vectors and in turn. These row vectors define the vertices of the unit square. With the more common matrix-vector product Ax, the parallelogram has vertices at and (note that Ax = (xTAT)T ). Thus when the determinant is equal to one, then the matrix represents an equi-areal mapping. The determinant of a matrix of arbitrary size can be defined by the Leibniz formula (as explained in the next paragraph) or the Laplace formula (as explained at the end of the Properties section). The Leibniz formula for the determinant of an n-by-n matrix A is Here the sum is computed over all permutations σ of the numbers {1, 2, ..., n}. A permutation is a function that reorders this set of integers. For example, for n = 3, the original sequence 1, 2, 3 might be reordered to 2, 3, 1 or 3, 2, 1. It is a basic fact of combinatorics that there are n! = 1 · 2 · 3 · ... · n (n factorial) such permutations. The set of all such permutations is denoted Sn. To any such permutation σ one attaches the signature σ, it is +1 for even and −1 for odd permutations. Evenness or oddness can be defined as follows: the permutation is even (odd) if the new sequence can be obtained by an even number (odd, respectively) of switches of adjacent numbers. For example, starting from 1, 2, 3 and switching once one gets 1, 3, 2, switching once more yields 3, 1, 2, and finally, after a total of three (an odd number) switches, one gets 3, 2, 1. Therefore this permutation is odd. The permutation 2, 3, 1 is even (1, 2, 3 → 2, 1, 3 → 2, 3, 1, two switches). In any of the n factorial summands, the term is a shorthand for the product over the indicated matrix entries, where i ranges from 1 to n, or equivalently: For example, for n = 4 and σ = (1, 4, 3, 2), sgn = -1 (one pair switch), and the matrix entries are A11; A24; A33; A42. For small matrices, one gets back the formulae given in the previous sections. The formal extension to arbitrary dimensions was made by Tullio Levi-Civita, see (LeviCivita symbol) using a pseudo-tensor symbol. An alternative, but equivalent definition of the determinant can be obtained by using the following theorem: Let Mn(K) denote the set of all matrices over the field K. There exists exactly one function with the two properties: F is alternating multilinear with regard to columns; F(I) = 1. One can then define the determinant as the unique function with the above properties.