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.
The solution set for the equations x − y = −1 and 3x + y = 9 is the single point (2, 3). A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5]
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 (See [2] or section 4.1. [3]) The inverse to a Manin matrix M can be defined by the standard formula: = (), where M adj is adjugate matrix given by the standard formula - its (i,j)-th element is the column-determinant of the (n − 1) × (n − 1) matrix that results from deleting row j and column i of M and multiplication by (-1 ...
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an m×n matrix, x designates a column vector (that is, n×1-matrix) of n variables x 1, x 2, ..., x n, and b is an m×1-column vector, then the matrix equation =