Cramer's rule
Introduction
Cramer's rule is a theorem in linear algebra, named after Gabriel Cramer, that provides 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 matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations.
Mathematical Background
Cramer's rule is applicable to systems of linear equations, which are collections of equations in which the variables only appear to the first power and are not multiplied together. Each equation in the system can be represented as a linear combination of vectors, leading to a matrix representation of the system of equations. The determinant of this matrix, if non-zero, provides key information about the system, including its solvability and the values of the variables.
Derivation of Cramer's Rule
The derivation of Cramer's rule begins with a system of linear equations represented in matrix form. The determinant of the coefficient matrix is calculated, and if it is non-zero, the system has a unique solution. The rule is then derived by replacing the column in the coefficient matrix corresponding to a given variable with the column of constants from the right-hand side of the equations, and calculating the determinant of the resulting matrix. The value of the variable is then given by the ratio of this determinant to the determinant of the original coefficient matrix.
Application of Cramer's Rule
Cramer's rule is used to solve systems of linear equations. It is particularly useful in theoretical mathematics, where explicit formulas are often required. However, in practical computations, it is rarely used, as it is less efficient than other methods, such as Gaussian elimination or LU decomposition.
Limitations and Criticisms
While Cramer's rule is mathematically elegant and can provide explicit formulas for solutions, it is not often used in practical computations due to its computational complexity. The rule requires the calculation of determinants, which is a computationally intensive process, especially for large matrices. As a result, other methods are often preferred for solving systems of linear equations.