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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 ...
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.
The number of distinct terms (including those with a zero coefficient) in an n-th degree equation in two variables is (n + 1)(n + 2) / 2.This is because the n-th degree terms are ,, …,, numbering n + 1 in total; the (n − 1) degree terms are ,, …,, numbering n in total; and so on through the first degree terms and , numbering 2 in total, and the single zero degree term (the constant).
Cramer's rule is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of two determinants. [9]
In this case, the solution is given by Cramer's rule: = () =,,, …, where is the matrix formed by replacing the -th column of by the column vector . This follows immediately by column expansion of the determinant, i.e.
A trace diagram representing the adjugate of a matrix. In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled by matrices. The simplest trace diagrams represent the trace and determinant of a matrix.
Consistency and independence of the equations in the set is established because the determinant of coefficients is non-zero, so a solution can be found using Cramer's rule. Using the examples from the subsection Elements of signal-flow graphs, we construct the graph In the figure, a signal-flow graph in this case.
A rule is a theorem that establishes a useful formula (e.g. Bayes' rule and Cramer's rule). A law or principle is a theorem with wide applicability (e.g. the law of large numbers, law of cosines, Kolmogorov's zero–one law, Harnack's principle, the least-upper-bound principle, and the pigeonhole principle). [g]