Formula for solving systems of linear equations
In linear algebra, Cramer's rule is 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 of matrices obtained from it by replacing one column by the column vector of right-sides of the equations. It is named after Gabriel Cramer, who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748, and possibly knew of it as early as 1729.
Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. In the case of n equations in n unknowns, it requires computation of "n" + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant. Cramer's rule can also be numerically unstable even for 2×2 systems. However, Cramer's rule can be implemented with the same complexity as Gaussian elimination, (consistently requires twice as many arithmetic operations and has the same numerical stability when the same permutation matrices are applied).
General case.
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows:
formula_1
where the "n" × "n" matrix A has a nonzero determinant, and the vector formula_2 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:
formula_3
where formula_4 is the matrix formed by replacing the i-th column of A by the column vector b.
A more general version of Cramer's rule considers the matrix equation
formula_5
where the "n" × "n" matrix A has a nonzero determinant, and X, B are "n" × "m" matrices. Given sequences formula_6 and formula_7, let formula_8 be the "k" × "k" submatrix of X with rows in formula_9 and columns in formula_10. Let formula_11 be the "n" × "n" matrix formed by replacing the formula_12 column of A by the formula_13 column of B, for all formula_14. Then
formula_15
In the case formula_16, this reduces to the normal Cramer's rule.
The rule holds for systems of equations with coefficients and unknowns in any field, not just in the real numbers.
Proof.
The proof for Cramer's rule uses the following properties of the determinants: linearity with respect to any given column and the fact that the determinant is zero whenever two columns are equal, which is implied by the property that the sign of the determinant flips if you switch two columns.
Fix the index "j" of a column, and consider that the entries of the other columns have fixed values. This makes the determinant a function of the entries of the jth column. Linearity with respect of this column means that this function has the form 
formula_17
where the formula_18 are coefficients that depend on the entries of A that are not in column j. So, one has 
formula_19
(Laplace expansion provides a formula for computing the formula_18 but their expression is not important here.) 
If the function formula_21 is applied to any "other" column "k" of A, then the result is the determinant of the matrix obtained from A by replacing column "j" by a copy of column "k", so the resulting determinant is 0 (the case of two equal columns).
Now consider a system of n linear equations in n unknowns formula_22, whose coefficient matrix is A, with det("A") assumed to be nonzero:
formula_23
If one combines these equations by taking "C"1,"j" times the first equation, plus "C"2,"j" times the second, and so forth until "C""n","j" times the last, then for every k the resulting coefficient of xk becomes
formula_24
So, all coefficients become zero, except the coefficient of formula_25 that becomes formula_26 Similarly, the constant coefficient becomes formula_27 and the resulting equation is thus
formula_28
which gives the value of formula_25 as
formula_30
As, by construction, the numerator is the determinant of the matrix obtained from A by replacing column "j" by b, we get the expression of Cramer's rule as a necessary condition for a solution. 
It remains to prove that these values for the unknowns form a solution. Let M be the "n" × "n" matrix that has the coefficients of formula_21 as jth row, for formula_32 (this is the adjugate matrix for A). Expressed in matrix terms, we have thus to prove that 
formula_33
is a solution; that is, that 
formula_34
For that, it suffices to prove that 
formula_35
where formula_36 is the identity matrix.
The above properties of the functions formula_21 show that one has "MA" 
 det("A")"In", and therefore,
formula_38
This completes the proof, since a left inverse of a square matrix is also a right-inverse (see Invertible matrix theorem).
For other proofs, see below.
Finding inverse matrix.
Let A be an "n" × "n" matrix with entries in a field "F". Then
formula_39
where adj("A") denotes the adjugate matrix, det("A") is the determinant, and "I" is the identity matrix. If det("A") is nonzero, then the inverse matrix of A is
formula_40
This gives a formula for the inverse of A, provided det("A") ≠ 0. In fact, this formula works whenever "F" is a commutative ring, provided that det("A") is a unit. If det("A") is not a unit, then A is not invertible over the ring (it may be invertible over a larger ring in which some non-unit elements of F may be invertible).
Applications.
Explicit formulas for small systems.
Consider the linear system
formula_41
which in matrix format is
formula_42
Assume "a"1"b"2 − "b"1"a"2 is nonzero. Then, with the help of determinants, x and y can be found with Cramer's rule as
formula_43
The rules for 3 × 3 matrices are similar. Given
formula_44
which in matrix format is
formula_45
Then the values of x, y and z can be found as follows:
formula_46
Differential geometry.
Ricci calculus.
Cramer's rule is used in the Ricci calculus in various calculations involving the Christoffel symbols of the first and second kind.
In particular, Cramer's rule can be used to prove that the divergence operator on a Riemannian manifold is invariant with respect to change of coordinates. We give a direct proof, suppressing the role of the Christoffel symbols.
Let formula_47 be a Riemannian manifold equipped with local coordinates formula_48. Let formula_49 be a vector field. We use the summation convention throughout.
Theorem.
"The "divergence" of formula_50,"
formula_51
"is invariant under change of coordinates."
Computing derivatives implicitly.
Consider the two equations formula_52 and formula_53. When "u" and "v" are independent variables, we can define formula_54 and formula_55
An equation for formula_56 can be found by applying Cramer's rule.
Integer programming.
Cramer's rule can be used to prove that an integer programming problem whose constraint matrix is totally unimodular and whose right-hand side is integer, has integer basic solutions. This makes the integer program substantially easier to solve.
Ordinary differential equations.
Cramer's rule is used to derive the general solution to an inhomogeneous linear differential equation by the method of variation of parameters.
Geometric interpretation.
Cramer's rule has a geometric interpretation that can be considered also a proof or simply giving insight about its geometric nature. These geometric arguments work in general and not only in the case of two equations with two unknowns presented here.
Given the system of equations
formula_57
it can be considered as an equation between vectors
formula_58
The area of the parallelogram determined by formula_59 and formula_60 is given by the determinant of the system of equations:
formula_61
In general, when there are more variables and equations, the determinant of n vectors of length n will give the "volume" of the "parallelepiped" determined by those vectors in the n-th dimensional Euclidean space.
Therefore, the area of the parallelogram determined by formula_62 and formula_60 has to be formula_64 times the area of the first one since one of the sides has been multiplied by this factor. Now, this last parallelogram, by Cavalieri's principle, has the same area as the parallelogram determined by formula_65 and formula_66
Equating the areas of this last and the second parallelogram gives the equation
formula_67
from which Cramer's rule follows.
Other proofs.
A proof by abstract linear algebra.
This is a restatement of the proof above in abstract language.
Consider the map formula_68 where formula_69 is the matrix formula_50 with formula_71 substituted in the formula_72th column, as in Cramer's rule. Because of linearity of determinant in every column, this map is linear. Observe that it sends the formula_72th column of formula_50 to the formula_72th basis vector formula_76 (with 1 in the formula_72th place), because determinant of a matrix with a repeated column is 0. So we have a linear map which agrees with the inverse of formula_50 on the column space; hence it agrees with formula_79 on the span of the column space. Since formula_50 is invertible, the column vectors span all of formula_81, so our map really is the inverse of formula_50. Cramer's rule follows.
A short proof.
A short proof of Cramer's rule can be given by noticing that formula_64 is the determinant of the matrix
formula_84
On the other hand, assuming that our original matrix A is invertible, this matrix formula_85 has columns formula_86, where formula_87 is the "n"-th column of the matrix A. Recall that the matrix formula_88 has columns formula_89, and therefore formula_90. Hence, by using that the determinant of the product of two matrices is the product of the determinants, we have
formula_91
The proof for other formula_25 is similar.
Inconsistent and indeterminate cases.
A system of equations is said to be inconsistent when there are no solutions and it is called indeterminate when there is more than one solution. For linear equations, an indeterminate system will have infinitely many solutions (if it is over an infinite field), since the solutions can be expressed in terms of one or more parameters that can take arbitrary values.
Cramer's rule applies to the case where the coefficient determinant is nonzero. In the 2×2 case, if the coefficient determinant is zero, then the system is incompatible if the numerator determinants are nonzero, or indeterminate if the numerator determinants are zero.
For 3×3 or higher systems, the only thing one can say when the coefficient determinant equals zero is that if any of the numerator determinants are nonzero, then the system must be inconsistent. However, having all determinants zero does not imply that the system is indeterminate. A simple example where all determinants vanish (equal zero) but the system is still incompatible is the 3×3 system "x"+"y"+"z"=1, "x"+"y"+"z"=2, "x"+"y"+"z"=3.
References.
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