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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 ...
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more
Download as PDF; Printable version; In other projects ... The vector equation is equivalent to a matrix equation of the form ... Cramer's rule is an explicit formula ...
Specifically, if A is TU and b is integral, then linear programs of forms like {,} or {} have integral optima, for any c. Hence if A is totally unimodular and b is integral, every extreme point of the feasible region (e.g. { x ∣ A x ≥ b } {\displaystyle \{x\mid Ax\geq b\}} ) is integral and thus the feasible region is an integral polyhedron.
The theorem can be read almost directly on the reduced row echelon form as follows. The rank of a matrix is the number of nonzero rows in its reduced row echelon form. If the ranks of the coefficient matrix and the augmented matrix are different, then the last non zero row has the form [ 0 … 0 ∣ 1 ] , {\displaystyle [0\ldots 0\mid 1 ...
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.
Multilinear algebra is the study of functions with multiple vector-valued arguments, with the functions being linear maps with respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants, inner and outer products, and dual spaces.
Coefficient matrices are used in algorithms such as Gaussian elimination and Cramer's rule to find solutions to the system. The leading entry (sometimes leading coefficient [ citation needed ] ) of a row in a matrix is the first nonzero entry in that row.