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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 an explicit formula for the solution of a system of linear equations, ... (if any exist) are given using the Moore–Penrose inverse 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.
Cramer's rule: In linear algebra, 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. Named after Swiss mathematician Gabriel Cramer .
An example of a degenerate case, in which n(n + 3) / 2 points on the curve are not sufficient to determine the curve uniquely, was provided by Cramer as part of Cramer's paradox. Let the degree be n = 3, and let nine points be all combinations of x = −1, 0, 1 and y = −1, 0, 1.
Rule of Sarrus: The determinant of the three columns on the left is the sum of the products along the down-right diagonals minus the sum of the products along the up-right diagonals. In matrix theory , the rule of Sarrus is a mnemonic device for computing the determinant of a 3 × 3 {\displaystyle 3\times 3} matrix named after the French ...
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...