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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 ...
The solution set for the equations x − y = −1 and 3x + y ... The system has a unique solution. ... Cramer's rule is an explicit formula for the solution of a ...
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 1750 he published Cramer's rule, giving a general formula for the solution for any unknown in a linear equation system having a unique solution, in terms of determinants implied by the system. This rule is still standard.
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 ...
Suppose is a directed graph without 2-dicycles, is the set of all dipaths in , and is the 0-1 incidence matrix of () versus . Then A {\displaystyle A} is totally unimodular if and only if every simple arbitrarily-oriented cycle in G {\displaystyle G} consists of alternating forwards and backwards arcs.
The chain rule has a particularly elegant statement in terms of total derivatives. It says that, for two functions f {\displaystyle f} and g {\displaystyle g} , the total derivative of the composite function f ∘ g {\displaystyle f\circ g} at a {\displaystyle a} satisfies
For most practical applications, it is not necessary to invert a matrix to solve a system of linear equations; however, for a unique solution, it is necessary that the matrix involved be invertible. Decomposition techniques like LU decomposition are much faster than inversion, and various fast algorithms for special classes of linear systems ...