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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 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).
Here is a theoretical question that Gaussian elimination cannot answer for you, but Cramer's rule can: suppose you have a system depending on (one or more) parameters that can vary continuously, and its coefficient matrix is always nonsingular, so that for all parameter values there is a unique solution.
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.
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Cramér's theorem may refer to . Cramér’s decomposition theorem, a statement about the sum of normal distributed random variable; Cramér's theorem (large deviations), a fundamental result in the theory of large deviations
As a result of the work of H. Maier on gaps between consecutive primes, the exact formulation of Cramér's conjecture has been called into question [...] It is still probably true that for every constant c > 2 {\displaystyle c>2} , there is a constant d > 0 {\displaystyle d>0} such that there is a prime between x {\displaystyle x} and x + d ...
In mathematics, the Cramér–Wold theorem [1] [2] or the Cramér–Wold device [3] [4] is a theorem in measure theory and which states that a Borel probability measure on is uniquely determined by the totality of its one-dimensional projections.