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In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A {\displaystyle A} is an n × n {\displaystyle n\times n} matrix, where a i j {\displaystyle a_{ij}} is the entry in the i {\displaystyle i} -th row and j {\displaystyle j} -th ...
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or | A |. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix.
In mathematics, Dodgson condensation or method of contractants is a method of computing the determinants of square matrices.It is named for its inventor, Charles Lutwidge Dodgson (better known by his pseudonym, as Lewis Carroll, the popular author), who discovered it in 1866. [1]
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
The Gram determinant is the squared volume of the parallelotope with a 1, ..., a k as edges. With these conditions a non-trivial cross product only exists: as a binary product in three and seven dimensions; as a product of n − 1 vectors in n ≥ 3 dimensions, being the Hodge dual of the exterior product of the vectors
Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n.A k × k minor of A, also called minor determinant of order k of A or, if m = n, the (n − k) th minor determinant of A (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns.
The Bombieri-Pila version of the determinant method would later be dubbed the real-analytic determinant method. Oscar Marmon generalized Bombieri and Pila's results in 2010. [2] Bombieri and Pila's result was novel because of its uniformity with respect to the polynomials defining the curves.
While the determinant can be computed in polynomial time by Gaussian elimination, it is generally believed that the permanent cannot be computed in polynomial time. In computational complexity theory , a theorem of Valiant states that computing permanents is #P-hard , and even #P-complete for matrices in which all entries are 0 or 1 Valiant ...