Search results
Results from the WOW.Com Content Network
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 ...
To describe a set with an infinite number of solutions, typically some of the variables are designated as free (or independent, or as parameters), meaning that they are allowed to take any value, while the remaining variables are dependent on the values of the free variables. For example, consider the following system:
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.
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.
In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. 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).
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.
For example, a 2,1 represents the element at the second row and first column of the matrix. In mathematics , a matrix ( pl. : matrices ) is a rectangular array or table of numbers , symbols , or expressions , with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.
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.