Cramérszabály

Cramérszabály, also known as Cramér's rule, is a theorem in linear algebra that provides an explicit formula for the solution of a system of linear equations. It is named after the Swedish mathematician Harald Cramér. The rule applies to systems of $n$ linear equations with $n$ unknowns that have a unique solution.

For a system of linear equations represented in matrix form as $Ax = b$, where $A$ is an $n \times n$ square matrix, $x$ is a column vector of variables, and $b$ is a column vector of constants, Cramér's rule states that if the determinant of matrix $A$, denoted as $\det(A)$, is non-zero, then the system has a unique solution. The value of each variable $x_i$ can be found by replacing the $i$-th column of matrix $A$ with the vector $b$ to form a new matrix $A_i$. The solution for $x_i$ is then given by the ratio of the determinant of $A_i$ to the determinant of $A$: $x_i = \frac{\det(A_i)}{\det(A)}$.

Cramér's rule is primarily of theoretical importance and is not typically used for solving large systems of equations in practice due to its computational inefficiency. The calculation of determinants for large matrices is computationally expensive. For practical purposes, methods like Gaussian elimination or LU decomposition are preferred for solving linear systems. However, Cramér's rule offers a clear and direct formula for understanding the relationship between the coefficients, constants, and the unique solution of a linear system. It also serves as a foundation for further theoretical developments in linear algebra and related fields.