Search results
Results from the WOW.Com Content Network
Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. [7] In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant.
In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N that is its inverse (these are equivalent under Cramer's rule).
Non-square matrices, i.e. m-by-n matrices for which m ≠ n, do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, (n ≤ m), then A has a left inverse, an n-by-m matrix B such that BA = I n.
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
Cramer's rule is a closed-form expression, in terms of determinants, of the solution of a system of n linear equations in n unknowns. Cramer's rule is useful for reasoning about the solution, but, except for n = 2 or 3 , it is rarely used for computing a solution, since Gaussian elimination is a faster algorithm.
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
In linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of a "rank-1 update" to a matrix whose inverse has previously been computed. [1] [2] [3] That is, given an invertible matrix and the outer product of vectors and , the formula cheaply computes an updated matrix inverse (+)).
An example of a degenerate case, in which n(n + 3) / 2 points on the curve are not sufficient to determine the curve uniquely, was provided by Cramer as part of Cramer's paradox. Let the degree be n = 3, and let nine points be all combinations of x = −1, 0, 1 and y = −1, 0, 1.