Computationally, it is generally inefficient and thus not used in practical applications which may involve many equations. However, it is of theoretical importance in that it gives an explicit expression for the solution of the system.
The system of equations is represented in matrix multiplication form as:
:
where the Square Matrix is invertible and the vector is the column vector of the variables: .
The theorem then states that:
:
where is the matrix formed by replacing the ''i''th column of by the column vector .
Let ''R'' be a Commutative Ring , ''A'' an ''n''×''n'' matrix with coefficients in ''R''. Then
:
where Adj(''A'') denotes the Adjugate of ''A'', det(''A'') is the determinant, and I is the Identity Matrix .
A good way to use Cramer's rule on a 2×2 matrix is to use this formula:
Given
: and
:,
which in matrix format is
:
x and y can be solved with Cramer's rule as:
:
:and
:
Cramer's rule is also extremely useful for solving problems in differential geometry. Consider the two equations and . When u and v are independent variables, we can define and .
Finding an equation for is a trivial application of Cramer's rule.
First, calculate the first derivatives of F, G, x and y.
:
:
:
:
Substituting dx, dy into dF and dG, we have:
:
:
Since u, v are both independent, the coefficients of du, dv must be zero. So we can write out equations for the coefficients:
:
:
:
:
Now, by Cramer's rule, we see that:
:
This is now a formula in terms of two Jacobian s:
:
Similar formulae can be derived for , , .
Cramer's rule can be used to prove the Cayley-Hamilton Theorem of linear algebra, as well as Nakayama's Lemma , which is fundamental in commutative ring theory.
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