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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 = 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]
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 Cramér–Rao bound is stated in this section for several increasingly general cases, beginning with the case in which the parameter is a scalar and its estimator is unbiased.
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).
Consider the system of equations x + y + 2z = 3, x + y + z = 1, 2x + 2y + 2z = 2.. The coefficient matrix is = [], and the augmented matrix is (|) = [].Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are infinitely many solutions.
Cramer's rule (See [2] or section 4.1. [3]) The inverse to a Manin matrix M can be defined by the standard formula: M − 1 = 1 det c o l ( M ) M a d j , {\displaystyle M^{-1}={\frac {1}{{\det }^{col}(M)}}M^{adj},} where M adj is adjugate matrix given by the standard formula - its (i,j)-th element is the column-determinant of the (n − 1) × ...
In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function.The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound, which may decay faster than exponential (e.g. sub-Gaussian).