Mathematics/Linear Algebra

Cramer’s Formula

Daniel-Kim 2011. 12. 21. 17:08

http://en.wikipedia.org/wiki/Cramer's_rule



Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows:

Ax = b\,
 (1)\,

where the n by n matrix A has a nonzero determinant, and the vector x = (x_1, \ldots, x_n)^\top is the column vector of the variables.

Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns are given by:

x_i = \frac{\det(A_i)}{\det(A)} \qquad i = 1, \ldots, n \,

where Ai is the matrix formed by replacing the ith column of A by the column vector b.

The rule holds for systems of equations with coefficients and unknowns in any field, not just in the real numbers. It has recently been shown that Cramer's rule can be implemented in O(n3) time,[3] which is comparable to more common methods of solving systems of linear equations, such as Gaussian elimination.

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Explicit formulas for small systems

Consider the linear system \left\{\begin{matrix}ax+by&={\color{red}e}\\ cx + dy&= {\color{red}f}\end{matrix}\right.\ which in matrix format is \begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} {\color{red}e} \\ {\color{red}f} \end{bmatrix}. Then, x and y can be found with Cramer's rule as

x = \begin{vmatrix} \color{red}{e} & b \\ \color{red}{f} & d \end{vmatrix}/\begin{vmatrix} a & b \\ c & d \end{vmatrix} = { {\color{red}e}d - b{\color{red}f} \over ad - bc} and
y = \begin{vmatrix} a & \color{red}{e} \\ c & \color{red}{f} \end{vmatrix}/\begin{vmatrix} a & b \\ c & d \end{vmatrix} = { a{\color{red}f} - {\color{red}e}c \over ad - bc}.

The rules for 3×3 are similar. Given \left\{\begin{matrix}ax + by + cz&= {\color{red}j}\\dx + ey + fz&= {\color{red}k}\\gx + hy + iz&= {\color{red}l}\end{matrix}\right. which in matrix format is

\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} {\color{red}j} \\ {\color{red}k} \\ {\color{red}l} \end{bmatrix} the values of x, y and z can be found as follows:
x = \frac { \begin{vmatrix} {\color{red}j} & b & c \\ {\color{red}k} & e & f \\ {\color{red}l} & h & i \end{vmatrix} } { \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} }, \quad y = \frac { \begin{vmatrix} a & {\color{red}j} & c \\ d & {\color{red}k} & f \\ g & {\color{red}l} & i \end{vmatrix} } { \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} },\text{ and }z = \frac { \begin{vmatrix} a & b & {\color{red}j} \\ d & e & {\color{red}k} \\ g & h & {\color{red}l} \end{vmatrix} } { \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} }.
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