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Cramer's rule. 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 ...
In matrix theory, the rule of Sarrus is a mnemonic device for computing the determinant of a matrix named after the French mathematician Pierre Frédéric Sarrus. [1] Consider a matrix. then its determinant can be computed by the following scheme. Write out the first two columns of the matrix to the right of the third column, giving five ...
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. [1][2] For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously ...
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.
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of ...
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
Rouché–Capelli theorem is a theorem in linear algebra that determines the number of solutions for a system of linear equations, given the rank of its augmented matrix and coefficient matrix. The theorem is variously known as the: Rouché–Capelli theorem in English speaking countries, Italy and Brazil; Kronecker–Capelli theorem in Austria ...
Invertible matrix. In linear algebra, an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. Invertible matrices are the same size as their inverse.