Cramer's Rule - Revision history

Khanh at 19:59, 29 January 2022

2022-01-29T19:59:14Z

Khanh: /* Geometric interpretation */

2022-01-29T19:57:58Z

Geometric interpretation

Khanh: /* Ordinary differential equations */

2022-01-29T19:57:21Z

Ordinary differential equations

Khanh: /* Integer programming */

2022-01-29T19:53:08Z

Integer programming

Khanh: /* Differential geometry */

2022-01-29T19:52:05Z

Differential geometry

Khanh: /* Proof */

2022-01-29T19:47:07Z

Proof

Khanh: Created page with "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 ha..."

2022-01-29T19:43:22Z

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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-hand-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750, (and possibly knew of it as early as 1729).

Cramer's rule implemented in a naïve way is computationally inefficient for systems of more than two or three equations. In the case of {{mvar|n}} equations in {{mvar|n}} unknowns, it requires computation of {{math|''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, it has recently been shown that Cramer's rule can be implemented in O(''n''3) time, which is comparable to more common methods of solving systems of linear equations, such as Gaussian elimination (consistently requiring 2.5 times as many arithmetic operations for all matrix sizes), while exhibiting comparable numeric stability in most cases.

==General case==
Consider a system of {{mvar|n}} linear equations for {{mvar|n}} unknowns, represented in matrix multiplication form as follows:

:<math> A\mathbf{x} = \mathbf{b}</math>

where the {{math|''n'' × ''n''}} matrix {{mvar|A}} has a nonzero determinant, and the vector <math> \mathbf{x} = (x_1, \ldots, x_n)^\mathsf{T} </math> 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:

:<math> x_i = \frac{\det(A_i)}{\det(A)} \qquad i = 1, \ldots, n</math>

where <math> A_i </math> is the matrix formed by replacing the {{mvar|i}}-th column of {{mvar|A}} by the column vector {{math|'''b'''}}.

A more general version of Cramer's rule considers the matrix equation

:<math> AX = B</math>

where the {{math|''n'' × ''n''}} matrix {{mvar|A}} has a nonzero determinant, and {{mvar|X}}, {{mvar|B}} are {{math|''n'' × ''m''}} matrices. Given sequences <math> 1 \leq i_1 < i_2 < \cdots < i_k \leq n </math> and <math> 1 \leq j_1 < j_2 < \cdots < j_k \leq m </math>, let <math> X_{I,J} </math> be the {{math|''k'' × ''k''}} submatrix of {{mvar|X}} with rows in <math> I := (i_1, \ldots, i_k ) </math> and columns in <math> J := (j_1, \ldots, j_k ) </math>. Let <math> A_{B}(I,J) </math> be the {{math|''n'' × ''n''}} matrix formed by replacing the <math>i_s</math> column of {{mvar|A}} by the <math>j_s</math> column of {{Mvar|B}}, for all <math> s = 1,\ldots, k </math>. Then

:<math> \det X_{I,J} = \frac{\det(A_{B}(I,J))}{\det(A)}. </math>

In the case <math> k = 1 </math>, 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. Linearity means that if we consider only column ''j'' as variable (fixing the others arbitrarily), the resulting function {{math|'''R'''''n'' → '''R'''}} (assuming matrix entries are in {{math|'''R'''}}) can be given by a matrix, with one row and ''n'' columns, that acts on column ''j''. In fact this is precisely what Laplace expansion does, writing {{math|1=det(''A'') = ''C''1''a''1,''j'' + ⋯ + ''Cnan,j''}} for certain coefficients ''C''1, ..., ''Cn'' that depend on the columns of {{mvar|A}} other than column ''j'' (the precise expression for these cofactors is not important here). The value {{math|det(''A'')}} is then the result of applying the one-line matrix {{math|1=''L''(''j'') = (''C''1 ''C''2 ⋯ ''Cn'')}} to column ''j'' of {{mvar|A}}. If {{math|''L''(''j'')}} is applied to any ''other'' column ''k'' of {{mvar|A}}, then the result is the determinant of the matrix obtained from {{mvar|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 {{mvar|n}} linear equations in {{mvar|n}} unknowns <math>x_1, \ldots,x_n</math>, whose coefficient matrix is {{mvar|A}}, with det(''A'') assumed to be nonzero:

:<math>\begin{matrix}
a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n&=&b_1\\
a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n&=&b_2\\
&\vdots&\\
a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n&=&b_n.
\end{matrix}</math>

If one combines these equations by taking ''C''1 times the first equation, plus ''C''2 times the second, and so forth until ''C''''n'' times the last, then the coefficient of {{mvar|xj}} will become {{math|1=''C''1''a''1, ''j'' + ⋯ + ''Cnan,j'' = det(''A'')}}, while the coefficients of all other unknowns become 0; the left hand side becomes simply det(''A'')''xj''. The right hand side is {{math|''C''1''b''1 + ⋯ + ''Cnbn''}}, which is {{math|''L''(''j'')}} applied to the column vector '''b''' of the right hand side {{mvar|bi}}. In fact what has been done here is multiply the matrix equation {{math|''A'''''x''' {{=}} '''b'''}} on the left by {{math|''L''(''j'')}}. Dividing by the nonzero number det(''A'') one finds the following equation, necessary to satisfy the system:

:<math>x_j=\frac{L_{(j)}\cdot\mathbf{b}}{\det(A)}.</math>

But by construction the numerator is the determinant of the matrix obtained from {{mvar|A}} by replacing column ''j'' by '''b''', so we get the expression of Cramer's rule as a necessary condition for a solution. The same procedure can be repeated for other values of ''j'' to find values for the other unknowns.

The only point that remains to prove is that these values for the unknowns, the only possible ones, do indeed together form a solution. But if the matrix {{mvar|A}} is invertible with inverse {{math|''A''−1}}, then {{math|'''x''' {{=}} ''A''−1'''b'''}} will be a solution, thus showing its existence. To see that {{mvar|A}} is invertible when det(''A'') is nonzero, consider the {{math|''n'' × ''n''}} matrix ''M'' obtained by stacking the one-line matrices {{math|''L''(''j'')}} on top of each other for ''j'' = 1, ..., ''n'' (this gives the [[adjugate matrix]] for {{mvar|A}}). It was shown that {{math|1=''L''(''j'')''A'' = (0 ⋯ 0 det(''A'') 0 ⋯ 0)}} where {{math|det(''A'')}} appears at the position ''j''; from this it follows that {{math|''MA'' {{=}} det(''A'')''In''}}. Therefore,

:<math>\frac1{\det(A)}M=A^{-1},</math>

completing the proof.

For other proofs, see below.

==Finding inverse matrix==
Let {{mvar|A}} be an {{math|''n'' × ''n''}} matrix with entries in a field {{math|''F''}}. Then

:<math>A\,\operatorname{adj}(A) = \operatorname{adj}(A)\,A=\det(A) I</math>

where {{math|adj(''A'')}} denotes the adjugate matrix, {{math|det(''A'')}} is the determinant, and {{math|''I''}} is the identity matrix. If {{math|det(''A'')}} is nonzero, then the inverse matrix of {{mvar|A}} is

:<math>A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A).</math>

This gives a formula for the inverse of {{mvar|A}}, provided {{math|det(''A'') ≠ 0}}. In fact, this formula works whenever {{math|''F''}} is a commutative ring, provided that {{math|det(''A'')}} is a unit. If {{math|det(''A'')}} is not a unit, then {{mvar|A}} is not invertible over the ring (it may be invertible over a larger ring in which some non-unit elements of {{mvar|F}} may be invertible).

==Applications==

===Explicit formulas for small systems===
Consider the linear system

:<math>\left\{\begin{matrix}
a_1x + b_1y&= {\color{red}c_1}\\
a_2x + b_2y&= {\color{red}c_2}
\end{matrix}\right.</math>

which in matrix format is

:<math>\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} {\color{red}c_1} \\ {\color{red}c_2} \end{bmatrix}.</math>

Assume {{math|''a''1''b''2 − ''b''1''a''2}} nonzero. Then, with help of determinants, {{mvar|x}} and {{mvar|y}} can be found with Cramer's rule as

:<math>\begin{align}
x &= \frac{\begin{vmatrix} {\color{red}{c_1}} & b_1 \\ {\color{red}{c_2}} & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}} = { {\color{red}c_1}b_2 - b_1{\color{red}c_2} \over a_1b_2 - b_1a_2}, \quad
y = \frac{\begin{vmatrix} a_1 & {\color{red}{c_1}} \\ a_2 & {\color{red}{c_2}} \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}} = { a_1{\color{red}c_2} - {\color{red}c_1}a_2 \over a_1b_2 - b_1a_2}
\end{align}.</math>

The rules for {{math|3 × 3}} matrices are similar. Given

:<math>\left\{\begin{matrix}
a_1x + b_1y + c_1z&= {\color{red}d_1}\\
a_2x + b_2y + c_2z&= {\color{red}d_2}\\
a_3x + b_3y + c_3z&= {\color{red}d_3}
\end{matrix}\right.</math>

which in matrix format is

:<math>\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} {\color{red}d_1} \\ {\color{red}d_2} \\ {\color{red}d_3} \end{bmatrix}.</math>

Then the values of {{mvar|x, y}} and {{mvar|z}} can be found as follows:

:<math>x = \frac{\begin{vmatrix} {\color{red}d_1} & b_1 & c_1 \\ {\color{red}d_2} & b_2 & c_2 \\ {\color{red}d_3} & b_3 & c_3 \end{vmatrix} } { \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}}, \quad
y = \frac {\begin{vmatrix} a_1 & {\color{red}d_1} & c_1 \\ a_2 & {\color{red}d_2} & c_2 \\ a_3 & {\color{red}d_3} & c_3 \end{vmatrix}} {\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}}, \text{ and }
z = \frac { \begin{vmatrix} a_1 & b_1 & {\color{red}d_1} \\ a_2 & b_2 & {\color{red}d_2} \\ a_3 & b_3 & {\color{red}d_3} \end{vmatrix}} {\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} }.</math>

===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 <math>(M,g)</math> be a Riemannian manifold equipped with local coordinates <math> (x^1, x^2, \dots, x^n)</math>. Let <math>A=A^i \frac{\partial}{\partial x^i}</math> be a vector field. We use the summation convention throughout.

:'''Theorem'''.
:''The ''divergence'' of <math>A</math>,
::<math> \operatorname{div} A = \frac{1}{\sqrt{\det g}} \frac{\partial}{\partial x^i} \left( A^i \sqrt{\det g} \right),</math>
:is invariant under change of coordinates.''

{{Collapse top|title=''Proof''}}
Let <math>(x^1,x^2,\ldots,x^n)\mapsto (\bar x^1,\ldots,\bar x^n)</math> be a coordinate transformation with [[invertible matrix|non-singular]] [[Jacobian matrix and determinant|Jacobian]]. Then the classical [[Vector field#Coordinate transformation law|transformation laws]] imply that <math>A=\bar A^{k}\frac{\partial}{\partial\bar x^{k}}</math> where <math>\bar A^{k}=\frac{\partial \bar x^{k}}{\partial x^{j}}A^{j}</math>. Similarly, if <math>g=g_{mk}\,dx^{m}\otimes dx^{k}=\bar{g}_{ij}\,d\bar x^{i}\otimes d\bar x^{j}</math>, then <math>\bar{g}_{ij}=\,\frac{\partial x^{m}}{\partial\bar x^{i}}\frac{\partial x^{k}}{\partial \bar x^{j}}g_{mk}</math>.
Writing this transformation law in terms of matrices yields <math>\bar g=\left(\frac{\partial x}{\partial\bar{x}}\right)^{\text{T}}g\left(\frac{\partial x}{\partial\bar{x}}\right)</math>, which implies <math>\det\bar g=\left(\det\left(\frac{\partial x}{\partial\bar{x}}\right)\right)^{2}\det g</math>.

Now one computes
:<math>\begin{align}
\operatorname{div} A &=\frac{1}{\sqrt{\det g}}\frac{\partial}{\partial x^{i}}\left( A^{i}\sqrt{\det g}\right)\\
&=\det\left(\frac{\partial x}{\partial\bar{x}}\right)\frac{1}{\sqrt{\det\bar g}}\frac{\partial \bar x^k}{\partial x^{i}}\frac{\partial}{\partial\bar x^{k}}\left(\frac{\partial x^{i}}{\partial \bar x^{\ell}}\bar{A}^{\ell}\det\!\left(\frac{\partial x}{\partial\bar{x}}\right)^{\!\!-1}\!\sqrt{\det\bar g}\right).
\end{align}</math>
In order to show that this equals
<math>\frac{1}{\sqrt{\det\bar g}}\frac{\partial}{\partial\bar x^{k}}\left(\bar A^{k}\sqrt{\det\bar{g}}\right)</math>,
it is necessary and sufficient to show that
:<math>\frac{\partial\bar x^{k}}{\partial x^{i}}\frac{\partial}{\partial\bar x^{k}}\left(\frac{\partial x^{i}}{\partial \bar x^{\ell}}\det\!\left(\frac{\partial x}{\partial\bar{x}}\right)^{\!\!\!-1}\right)=0\qquad\text{for all } \ell, </math>
which is equivalent to
:<math>\frac{\partial}{\partial \bar x^{\ell}}\det\left(\frac{\partial x}{\partial\bar{x}}\right)
=\det\left(\frac{\partial x}{\partial\bar{x}}\right)\frac{\partial\bar x^{k}}{\partial x^{i}}\frac{\partial^{2}x^{i}}{\partial\bar x^{k}\partial\bar x^{\ell}}.
</math>
Carrying out the differentiation on the left-hand side, we get:
:<math>\begin{align}
\frac{\partial}{\partial\bar x^{\ell}}\det\left(\frac{\partial x}{\partial\bar{x}}\right)
&=(-1)^{i+j}\frac{\partial^{2}x^{i}}{\partial\bar x^{\ell}\partial\bar x^{j}}\det M(i|j)\\
&=\frac{\partial^{2}x^{i}}{\partial\bar x^{\ell}\partial\bar x^{j}}\det\left(\frac{\partial x}{\partial\bar{x}}\right)\frac{(-1)^{i+j}}{\det\left(\frac{\partial x}{\partial\bar{x}}\right)}\det M(i|j)=(\ast),
\end{align}</math>
where <math>M(i|j)</math> denotes the matrix obtained from <math>\left(\frac{\partial x}{\partial\bar{x}}\right)</math> by deleting the <math>i</math>th row and <math>j</math>th column.
But Cramer's Rule says that
:<math>\frac{(-1)^{i+j}}{\det\left(\frac{\partial x}{\partial\bar{x}}\right)}\det M(i|j) </math>
is the <math>(j,i)</math>th entry of the matrix <math>\left(\frac{\partial \bar{x}}{\partial x}\right)</math>.
Thus
:<math>(\ast)=\det\left(\frac{\partial x}{\partial\bar{x}}\right)\frac{\partial^{2}x^{i}}{\partial\bar x^{\ell}\partial\bar x^{j}}\frac{\partial\bar x^{j}}{\partial x^{i}},</math>
completing the proof.

{{Collapse bottom}}

====Computing derivatives implicitly====
Consider the two equations <math>F(x, y, u, v) = 0</math> and <math>G(x, y, u, v) = 0</math>. When ''u'' and ''v'' are independent variables, we can define <math>x = X(u, v)</math> and <math>y = Y(u, v).</math>

An equation for <math>\dfrac{\partial x}{\partial u}</math> can be found by applying Cramer's rule.

{{Collapse top|title=''Calculation of <math>\dfrac{\partial x}{\partial u}</math>''}}
First, calculate the first derivatives of ''F'', ''G'', ''x'', and ''y'':

:<math>\begin{align}
dF &= \frac{\partial F}{\partial x} dx + \frac{\partial F}{\partial y} dy +\frac{\partial F}{\partial u} du +\frac{\partial F}{\partial v} dv = 0 \\[6pt]
dG &= \frac{\partial G}{\partial x} dx + \frac{\partial G}{\partial y} dy +\frac{\partial G}{\partial u} du +\frac{\partial G}{\partial v} dv = 0 \\[6pt]
dx &= \frac{\partial X}{\partial u} du + \frac{\partial X}{\partial v} dv \\[6pt]
dy &= \frac{\partial Y}{\partial u} du + \frac{\partial Y}{\partial v} dv.
\end{align}</math>

Substituting ''dx'', ''dy'' into ''dF'' and ''dG'', we have:

:<math>\begin{align}
dF &= \left(\frac{\partial F}{\partial x} \frac{\partial x}{\partial u} +\frac{\partial F}{\partial y} \frac{\partial y}{\partial u} + \frac{\partial F}{\partial u} \right) du + \left(\frac{\partial F}{\partial x} \frac{\partial x}{\partial v} +\frac{\partial F}{\partial y} \frac{\partial y}{\partial v} +\frac{\partial F}{\partial v} \right) dv = 0 \\ [6pt]
dG &= \left(\frac{\partial G}{\partial x} \frac{\partial x}{\partial u} +\frac{\partial G}{\partial y} \frac{\partial y}{\partial u} +\frac{\partial G}{\partial u} \right) du + \left(\frac{\partial G}{\partial x} \frac{\partial x}{\partial v} +\frac{\partial G}{\partial y} \frac{\partial y}{\partial v} +\frac{\partial G}{\partial v} \right) dv = 0.
\end{align}</math>

Since ''u'', ''v'' are both independent, the coefficients of ''du'', ''dv'' must be zero. So we can write out equations for the coefficients:

:<math>\begin{align}
\frac{\partial F}{\partial x} \frac{\partial x}{\partial u} +\frac{\partial F}{\partial y} \frac{\partial y}{\partial u} & = -\frac{\partial F}{\partial u} \\[6pt]
\frac{\partial G}{\partial x} \frac{\partial x}{\partial u} +\frac{\partial G}{\partial y} \frac{\partial y}{\partial u} & = -\frac{\partial G}{\partial u} \\[6pt]
\frac{\partial F}{\partial x} \frac{\partial x}{\partial v} +\frac{\partial F}{\partial y} \frac{\partial y}{\partial v} & = -\frac{\partial F}{\partial v} \\[6pt]
\frac{\partial G}{\partial x} \frac{\partial x}{\partial v} +\frac{\partial G}{\partial y} \frac{\partial y}{\partial v} & = -\frac{\partial G}{\partial v}.
\end{align}</math>

Now, by Cramer's rule, we see that:

:<math>\frac{\partial x}{\partial u} = \frac{\begin{vmatrix} -\frac{\partial F}{\partial u} & \frac{\partial F}{\partial y} \\ -\frac{\partial G}{\partial u} & \frac{\partial G}{\partial y}\end{vmatrix}}{\begin{vmatrix}\frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} \\ \frac{\partial G}{\partial x} & \frac{\partial G}{\partial y}\end{vmatrix}}.</math>

This is now a formula in terms of two Jacobians:

:<math>\frac{\partial x}{\partial u} = -\frac{\left(\frac{\partial (F, G)}{\partial (u, y)}\right)}{\left(\frac{\partial (F, G)}{\partial(x, y)}\right)}.</math>

Similar formulas can be derived for <math>\frac{\partial x}{\partial v}, \frac{\partial y}{\partial u}, \frac{\partial y}{\partial v}.</math>
{{Collapse bottom}}

===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==
[[File:Cramer.jpg|thumb|400px|Geometric interpretation of Cramer's rule. The areas of the second and third shaded parallelograms are the same and the second is <math>x_1</math> times the first. From this equality Cramer's rule follows.]]
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

:<math>\begin{matrix}a_{11}x_1+a_{12}x_2&=b_1\\a_{21}x_1+a_{22}x_2&=b_2\end{matrix}</math>

it can be considered as an equation between vectors

:<math>x_1\binom{a_{11}}{a_{21}}+x_2\binom{a_{12}}{a_{22}}=\binom{b_1}{b_2}. </math>

The area of the parallelogram determined by <math>\binom{a_{11}}{a_{21}}</math> and <math>\binom{a_{12}}{a_{22}}</math> is given by the determinant of the system of equations:

:<math>\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}.</math>

In general, when there are more variables and equations, the determinant of {{mvar|n}} vectors of length {{mvar|n}} will give the ''volume'' of the ''[[parallelepiped]]'' determined by those vectors in the {{mvar|n}}-th dimensional Euclidean space.

Therefore, the area of the parallelogram determined by <math>x_1\binom{a_{11}}{a_{21}}</math> and <math>\binom{a_{12}}{a_{22}}</math> has to be <math>x_1</math> 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 <math>\binom{b_1}{b_2}=x_1\binom{a_{11}}{a_{21}}+x_2\binom{a_{12}}{a_{22}}</math> and <math>\binom{a_{12}}{a_{22}}.</math>

Equating the areas of this last and the second parallelogram gives the equation

:<math>\begin{vmatrix}b_1&a_{12}\\b_2&a_{22}\end{vmatrix} = \begin{vmatrix}a_{11}x_1&a_{12}\\a_{21}x_1&a_{22}\end{vmatrix} =x_1 \begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix} </math>

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 <math>\mathbf{x}=(x_1,\ldots, x_n) \mapsto \frac{1}{\det A} \left(\det (A_1),\ldots, \det(A_n)\right),</math> where <math>A_i</math> is the matrix <math>A</math> with <math>\mathbf{x}</math> substituted in the <math>i</math>th column, as in Cramer's rule. Because of linearity of determinant in every column, this map is linear. Observe that it sends the <math>i</math>th column of <math>A</math> to the <math>i</math>th basis vector <math>\mathbf{e}_i=(0,\ldots, 1, \ldots, 0) </math> (with 1 in the <math>i</math>th place), because determinant of a matrix with a repeated column is 0. So we have a linear map which agrees with the inverse of <math>A</math> on the column space; hence it agrees with <math>A^{-1}</math> on the span of the column space. Since <math>A</math> is invertible, the column vectors span all of <math>\mathbb{R}^n</math>, so our map really is the inverse of <math>A</math>. Cramer's rule follows.

===A short proof===
A short proof of Cramer's rule can be given by noticing that <math>x_1</math> is the determinant of the matrix

:<math>X_1=\begin{bmatrix}
x_1 & 0 & 0 & \cdots & 0\\
x_2 & 1 & 0 & \cdots & 0\\
x_3 & 0 & 1 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots &\vdots \\
x_n & 0 & 0 & \cdots & 1
\end{bmatrix}</math>

On the other hand, assuming that our original matrix {{mvar|A}} is invertible, this matrix <math>X_1</math> has columns <math>A^{-1}\mathbf{b}, A^{-1}\mathbf{v}_2, \ldots, A^{-1}\mathbf{v}_n </math>, where <math>\mathbf{v}_n</math> is the ''n''-th column of the matrix {{mvar|A}}. Recall that the matrix <math>A_1</math> has columns <math>\mathbf{b}, \mathbf{v}_2, \ldots, \mathbf{v}_n </math>, and therefore <math>X_1=A^{-1}A_1</math>. Hence, by using that the determinant of the product of two matrices is the product of the determinants, we have

:<math> x_1= \det (X_1) = \det (A^{-1}) \det (A_1)= \frac{\det (A_1)}{\det (A)}.</math>

The proof for other <math>x_j</math> is similar.

==Incompatible and indeterminate cases==
A system of equations is said to be incompatible or 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 incompatible. 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.

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	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 incompatible. 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.		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 incompatible. 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.
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikipedia.org/wiki/Cramer%27s_rule Cramer's rule, Wikipedia] under a CC BY-SA license

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 :<math>\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix}.</math>
-In general, when there are more variables and equations, the determinant of {{mvar|n}} vectors of length {{mvar|n}} will give the ''volume'' of the ''[[parallelepiped]]'' determined by those vectors in the {{mvar|n}}-th dimensional Euclidean space.
+In general, when there are more variables and equations, the determinant of {{mvar|n}} vectors of length {{mvar|n}} will give the ''volume'' of the ''parallelepiped'' determined by those vectors in the {{mvar|n}}-th dimensional Euclidean space.
 Therefore, the area of the parallelogram determined by <math>x_1\binom{a_{11}}{a_{21}}</math> and <math>\binom{a_{12}}{a_{22}}</math> has to be <math>x_1</math> 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 <math>\binom{b_1}{b_2}=x_1\binom{a_{11}}{a_{21}}+x_2\binom{a_{12}}{a_{22}}</math> and <math>\binom{a_{12}}{a_{22}}.</math>

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 ===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]].
+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==

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 ===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.
+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===

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 :is invariant under change of coordinates.''
-{{Collapse top|title=''Proof''}}
+<div style="text-align: center;">'''''Proof'''''</div>
-Let <math>(x^1,x^2,\ldots,x^n)\mapsto (\bar x^1,\ldots,\bar x^n)</math> be a coordinate transformation with [[invertible matrix|non-singular]] [[Jacobian matrix and determinant|Jacobian]].  Then the classical [[Vector field#Coordinate transformation law|transformation laws]] imply that <math>A=\bar A^{k}\frac{\partial}{\partial\bar x^{k}}</math> where <math>\bar A^{k}=\frac{\partial \bar x^{k}}{\partial x^{j}}A^{j}</math>.  Similarly, if <math>g=g_{mk}\,dx^{m}\otimes dx^{k}=\bar{g}_{ij}\,d\bar x^{i}\otimes d\bar x^{j}</math>, then <math>\bar{g}_{ij}=\,\frac{\partial x^{m}}{\partial\bar x^{i}}\frac{\partial x^{k}}{\partial \bar x^{j}}g_{mk}</math>.
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 Writing this transformation law in terms of matrices yields <math>\bar g=\left(\frac{\partial x}{\partial\bar{x}}\right)^{\text{T}}g\left(\frac{\partial x}{\partial\bar{x}}\right)</math>, which implies <math>\det\bar g=\left(\det\left(\frac{\partial x}{\partial\bar{x}}\right)\right)^{2}\det g</math>.
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 completing the proof.
-{{Collapse bottom}}
+</blockquote>
 ====Computing derivatives implicitly====
 Consider the two equations <math>F(x, y, u, v) = 0</math> and <math>G(x, y, u, v) = 0</math>.  When ''u'' and ''v'' are independent variables, we can define <math>x = X(u, v)</math> and <math>y = Y(u, v).</math>
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 An equation for <math>\dfrac{\partial x}{\partial u}</math> can be found by applying Cramer's rule.
-{{Collapse top|title=''Calculation of <math>\dfrac{\partial x}{\partial u}</math>''}}
+<div style="text-align: center;">'''''Calculation of <math>\dfrac{\partial x}{\partial u}</math>'''''</div>
 First, calculate the first derivatives of ''F'', ''G'', ''x'', and ''y'':
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 Similar formulas can be derived for <math>\frac{\partial x}{\partial v}, \frac{\partial y}{\partial u}, \frac{\partial y}{\partial v}.</math>
-{{Collapse bottom}}
+</blockquote>
 ===Integer programming===

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 \end{matrix}</math>
-If one combines these equations by taking ''C''<sub>1</sub> times the first equation, plus ''C''<sub>2</sub> times the second, and so forth until ''C''<sub>''n''</sub> times the last, then the coefficient of {{mvar|x<sub>j</sub>}} will become {{math|1=''C''<sub>1</sub>''a''<sub>1, ''j''</sub> + ⋯ + ''C<sub>n</sub>a<sub>n,j</sub>'' = det(''A'')}}, while the coefficients of all other unknowns become 0; the left hand side becomes simply det(''A'')''x<sub>j</sub>''. The right hand side is {{math|''C''<sub>1</sub>''b''<sub>1</sub> + ⋯ + ''C<sub>n</sub>b<sub>n</sub>''}}, which is {{math|''L''<sub>(''j'')</sub>}} applied to the column vector '''b''' of the right hand side {{mvar|b<sub>i</sub>}}. In fact what has been done here is multiply the matrix equation {{math|''A'''''x''' {{=}} '''b'''}} on the left by {{math|''L''<sub>(''j'')</sub>}}. Dividing by the nonzero number det(''A'') one finds the following equation, necessary to satisfy the system:
+If one combines these equations by taking ''C''<sub>1</sub> times the first equation, plus ''C''<sub>2</sub> times the second, and so forth until ''C''<sub>''n''</sub> times the last, then the coefficient of {{mvar|x<sub>j</sub>}} will become {{math|1=''C''<sub>1</sub>''a''<sub>1, ''j''</sub> + ⋯ + ''C<sub>n</sub>a<sub>n,j</sub>'' = det(''A'')}}, while the coefficients of all other unknowns become 0; the left hand side becomes simply det(''A'')''x<sub>j</sub>''. The right hand side is {{math|''C''<sub>1</sub>''b''<sub>1</sub> + ⋯ + ''C<sub>n</sub>b<sub>n</sub>''}}, which is {{math|''L''<sub>(''j'')</sub>}} applied to the column vector '''b''' of the right hand side {{mvar|b<sub>i</sub>}}. In fact what has been done here is multiply the matrix equation ''A'''''x''' = '''b''' on the left by {{math|''L''<sub>(''j'')</sub>}}. Dividing by the nonzero number det(''A'') one finds the following equation, necessary to satisfy the system:
 :<math>x_j=\frac{L_{(j)}\cdot\mathbf{b}}{\det(A)}.</math>
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 But by construction the numerator is the determinant of the matrix obtained from {{mvar|A}} by replacing column ''j'' by '''b''', so we get the expression of Cramer's rule as a necessary condition for a solution. The same procedure can be repeated for other values of ''j'' to find values for the other unknowns.
-The only point that remains to prove is that these values for the unknowns, the only possible ones, do indeed together form a solution. But if the matrix {{mvar|A}} is invertible with inverse {{math|''A''<sup>−1</sup>}}, then {{math|'''x''' {{=}} ''A''<sup>−1</sup>'''b'''}} will be a solution, thus showing its existence. To see that {{mvar|A}} is invertible when det(''A'') is nonzero, consider the {{math|''n'' × ''n''}} matrix ''M'' obtained by stacking the one-line matrices {{math|''L''<sub>(''j'')</sub>}} on top of each other for ''j'' = 1, ..., ''n'' (this gives the [[adjugate matrix]] for {{mvar|A}}). It was shown that {{math|1=''L''<sub>(''j'')</sub>''A'' = (0 ⋯ 0 det(''A'') 0 ⋯ 0)}} where {{math|det(''A'')}} appears at the position ''j''; from this it follows that {{math|''MA'' {{=}} det(''A'')''I<sub>n</sub>''}}. Therefore,
+The only point that remains to prove is that these values for the unknowns, the only possible ones, do indeed together form a solution. But if the matrix {{mvar|A}} is invertible with inverse {{math|''A''<sup>−1</sup>}}, then '''x''' = ''A''<sup>−1</sup>'''b''' will be a solution, thus showing its existence. To see that {{mvar|A}} is invertible when det(''A'') is nonzero, consider the {{math|''n'' × ''n''}} matrix ''M'' obtained by stacking the one-line matrices {{math|''L''<sub>(''j'')</sub>}} on top of each other for ''j'' = 1, ..., ''n'' (this gives the adjugate matrix for {{mvar|A}}). It was shown that {{math|1=''L''<sub>(''j'')</sub>''A'' = (0 ⋯ 0 det(''A'') 0 ⋯ 0)}} where {{math|det(''A'')}} appears at the position ''j''; from this it follows that ''MA'' = det(''A'')''I<sub>n</sub>''. Therefore,
 :<math>\frac1{\det(A)}M=A^{-1},</math>