General Case
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows:
|
|
where the n by n matrix has a nonzero determinant, and the vector 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:
where is the matrix formed by replacing the ith column of by the column vector .
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, which is comparable to more common methods of solving systems of linear equations, such as Gaussian elimination.
Read more about this topic: Cramer's Rule
Famous quotes containing the words general and/or case:
“A writer who writes, “I am alone” ... can be considered rather comical. It is comical for a man to recognize his solitude by addressing a reader and by using methods that prevent the individual from being alone. The word alone is just as general as the word bread. To pronounce it is to summon to oneself the presence of everything the word excludes.”
—Maurice Blanchot (b. 1907)
“If you have any information or evidence regarding the O.J. Simpson case, press 2 now. If you are an expert in fields relating to the O.J. Simpson case and would like to offer your services, press 3 now. If you would like the address where you can send a letter of support to O.J. Simpson, press 1 now. If you are seeking legal representation from the law offices of Robert L. Shapiro, press 4 now.”
—Advertisement. Aired August 8, 1994 by Tom Snyder on TV station CNBC. Chicago Sun Times, p. 11 (July 24, 1994)