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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 ...
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). Thus every equation Mx = b , where M and b both have integer components and M is unimodular, has an integer 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.
Inverse proportionality with product x y = 1 . Two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion) [2] if each of the variables is directly proportional to the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant. [3]
L'Hôpital's rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Lamarck's theory of evolution has two laws: The first can be paraphrased as "use it or lose it". The second is the more famous law of soft inheritance.
with probability density function (;) which satisfies the two regularity conditions below. The Fisher information matrix is a d × d {\displaystyle d\times d} matrix with element I m , k {\displaystyle I_{m,k}} defined as
For normally distributed random variables inverse-variance weighted averages can also be derived as the maximum likelihood estimate for the true value. Furthermore, from a Bayesian perspective the posterior distribution for the true value given normally distributed observations and a flat prior is a normal distribution with the inverse-variance weighted average as a mean and variance ().
It is in an odd place. But my guess is that the majority of (at least American) college studends first last and only interaction w/ Cramer's rule will be as a means to find the inverse to 2x2 matrices. It doesn't belong in a lvl2 heading though. Protonk 19:02, 17 July 2010 (UTC)