Cramer's Rule - Wikipedia, The Free Encyclopedia

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In linear algebra, Cramer's ruleis 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 vector of right hand sides of the equations. It is named after Gabriel Cramer (17041752), who published the

rule for an arbitrary number of unknowns in 1750,[1]although Colin Maclaurin also published special cases of the

rule in 1748[2](and possibly knew of it as early as 1729).[3][4][5]

1 General case

2 Proof

3 Finding inverse matrix

4 Applications

4.1 Explicit formulas for small systems

4.2 Differential geometry

4.3 Integer programming

4.4 Ordinary differential equations

5 Geometric interpretation

6 A short proof

6.1 Proof using Clifford algebra

7 Systems of vector equations: Cramers Rule extended.

7.1 Solving for unknown vectors.

7.2 Solving for unknown scalars.

7.3 Projecting a vector onto an arbitrary basis.

7.4 Projecting a vector onto an orthogonal basis.

7.5 Solving a system of vector equations using SymPy.

8 Incompatible and indeterminate cases

9 See also

10 Notes

11 External links

Consider a system of nlinear equations for nunknowns, represented in matrix multiplication form as follows:

er's rule - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Cram

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where the nby nmatrix 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,[6]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 mostcases).

The proof for Cramer's rule uses just two properties of determinants: linearity with respect to any given column

(taking for that column a linear combination of column vectors produces as determinant the corresponding linear

combination of their determinants), and the fact that the determinant is zero whenever two columns are equal

(which is implied by the basic property that the determinant is alternating in the columns).

Fix the indexjof a column. Linearity means that if we consider only columnjas variable (fixing the others

arbitrarily), the resulting function RnR(assuming matrix entries are in R) can be given by a matrix, with onerow and ncolumns, that acts on columnj. In fact this is precisely what Laplace expansion does, writing

det(A) = C1a

1,j+ + C

na

n,jfor certain coefficients C

1,,C

nthat depend on the columns ofAother than

columnj(the precise expression for these cofactors is not important here). The value det(A) is then the result of

applying the one-line matrixL(j)

= (C1C

2 C

n)to columnjofA. IfL

(j)is applied to any othercolumn kof

A, then the result is the determinant of the matrix obtained fromAby replacing columnjby a copy of column k,

so the resulting determinant is 0 (the case of two equal columns).

Now consider a system of nlinear equations in nunknowns , whose coefficient matrix isA, with

det(A) assumed to be nonzero:

If one combines these equations by taking C1times the first equation, plus C

2times the second, and so forth until


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Cntimes the last, then the coefficient ofx

jwill become C

1a

1,j+ + C

na

n,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

C1b

1+ + C

nb

n, which isL

(j)applied to the column vector bof the right hand sides b

i. In fact what has

been done here is multiply the matrix equationAx= bon the left byL(j)

. Dividing by the nonzero number

det(A) one finds the following equation, necessary to satisfy the system:

But by construction the numerator is the determinant of the matrix obtained fromAby replacing columnjby 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 ofjto 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 matrixAis invertible with inverseA1, then x=A1bwill be a solution,thus showing its existence. To see thatAis invertible when det(A) is nonzero, consider the nby nmatrixM

obtained by stacking the one-line matricesL(j)

on top of each other forj= 1, 2, , n(this gives the adjugate

matrix forA). It was shown thatL(j)

A= (0 0 det(A) 0 0)where det(A)appears at the positionj;

from this it follows thatMA= det(A)In

. Therefore

completing the proof.

LetAbe an nnmatrix. Then

where Adj(A) denotes the adjugate matrix ofA, det(A) is the determinant, andIis the identity matrix. If det(A) is

invertible inR, then the inverse matrix ofAis

IfRis a field (such as the field of real numbers), then this gives a formula for the inverse ofA, provided

det(A) 0. In fact, this formula will work wheneverRis a commutative ring, provided that det(A) is a unit. If

det(A) is not a unit, thenAis not invertible.

Explicit formulas for small systems


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Consider the linear system which in matrix format is

Assume adbcnonzero. Then,xandycan be found with Cramer's rule as

and

The rules for 33 are similar. Given which in matrix format is

Then the values ofx,yandzcan be found as follows:

Differential geometry

Cramer's rule is also extremely useful for solving problems in differential geometry. Consider the two equations

and . When uand vare 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, andy:


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Substituting dx, dyinto dFand dG, we have:

Since u, vare both independent, the coefficients of du, dvmust 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 Jacobians:


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Similar formulae can be derived for , ,

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.

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 twounknowns presented here.

Given the system of equations

it can be considered as an equation between vectors

The area of the parallelogram determined by and is given by the determinant of the system of

equations:

In general, when there are more variables and equations, the determinant of vectors of length will give the

volumeof theparallelepipeddetermined by those vectors in the -th dimensional Euclidean space.

Therefore the area of the parallelogram determined by and has to be 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 and .

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


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Geometric interpretation of Cramer's rule. The areas of the second

and third shaded parallelograms are the same and the second is

times the first. From this equality Cramer's rule follows.

from which Cramer's rule follows.


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A short proof of Cramer's rule [7]can be given by noticing that is the determinant of the matrix

On the other hand, assuming that our original matrix is invertible, this matrix has columns

, where is the th column of the matrix . Recall that the matrix has

columns . Hence we have , as wanted.

The proof for other is similar.

Proof using Clifford algebra

Consider the system of three scalar equations in three unknown scalars

and assign an orthonormal vector basis for as

Let the vectors

Adding the system of equations, it is seen that

Using outer products, each unknown scalar can be solved as


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For equations in unknowns, the solution for the th unknown generalizes to

If the are linearly independent, then the can be expressed in determinant form identical to Cramers Ruleas


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where denotes the substitution of vector with vector in the th numerator position.

Consider the system of vector equations in unknown vectors

where we want to solve for each unknown vector in terms of the given scalar constants and vector

constants .


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Solving for unknown vectors.

Using the Clifford geometric algebra of Euclidean vectors, the vectors and are in a vector space having

dimensions spanned by a basis of orthonormal base vectors . This -dimensional space

can be extended to be a subspace of a larger -dimensional space

.

Multiply the th equation by the th orthonormal base unit , using outer product on the right, as

The original system of equations in grade- vectors is now transformed into a system of equations in grade-

vectors, and no parallel components have been deleted by the outer products since they multiply on perpendicular

extended base units.

Let the vectors

Adding the transformed system of equations gives

which is a -vector equation. These outer (wedge) products are equal to geometric products since the factors are

perpendicular.

For , , , and are solved by multiplying , , and , respectively, on the

right with outer products


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In the solution of , and similarly for and , is a -blade having of its dimensions in the

extended dimensions , and the remaining one dimension is in the solution space of the vectors and .

The -blade is in the problem space, or the extended dimensions. The inner product

reduces, or contracts, to a -vector in the -dimensional solution space.

The divisor , the square of a blade, is a scalar product that can be computed by a determinant.

Since is a -vector, it commutes with the vectors without sign change and isconveniently shifted into the vacant th spot. A sign change occurs in every even th solution ,

such as , due to commuting or shifting right an odd number of times, in the dividend blade

, into its th spot.

In general, is solved as

where denotes replacing the th element with . The factor accounts for shifting the th

vector by places. The -blade is multiplied by inner product

with the reversed -blade , producing a -vector in the -dimensional solution

space.

Using this formula, for solving a system of vector equations having unknown vectors

in a -dimensional space, requires extending the space to dimensions. The extended dimensions are


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essentially used to hold the system of equations represented by the scalar constants -vectors and the vector

constants -vectors . The vector constants are grade-increased to -vectors or grade- vectors

that are partly in the extended space. Notice the similarity of form to Cramers Rule for

systems of scalar equations; a basis is added in both cases. The advantage of this formula is that it avoids scalar

coordinates and the results are directly in terms of vectors.

The system of vector equations can also be solved in terms of coordinates, without using the geometric algebra

formula above, by the usual process of expanding all the vectors in the system into their coordinate vector

components. In each expanded equation, the parallel (like) components are summed into groups that formindependent systems of unknown coordinates in equations. Each system solves for one dimension of

coordinates. After solving the systems, the solved vectors can be reassembled from the solved coordinates. It

seems that few books explicitly discuss this process for systems of vector equations. This process is the

application of the abstract concept of linear independence as it applies to linearly independent dimensions of

vector components or unit vectors. The linear independence concept extends to multivectors in geometric algebra,

where each unique unit blade is linearly independent of the others for the purpose of solving equations or systems

of equations. An equation containing a sum of linearly independent terms can be rewritten as separate

independent equations, each in the terms of one dimension.

Solving for unknown scalars.

It is also noticed that, instead of solving for unknown vectors , the maybe known vectors and the vectors

maybe unknown. The vectors , , and couldbe solved as

In general, vector maybe solved as


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and represents transforming or projecting the system, or each vector , onto the basis of vectors

which need not be orthonormal.However, solving for the vectors by this formula isunnecessary, and unnecessarily requires vectors at a time. Solving each equation is

independent in this case. This has been shown to clarify the usage, as far as what not to do, unless one has an

unusual need to solve a particular vector . Instead, the following can be done in the case of projecting vectors

onto a new arbitrary basis .

Projecting a vector onto an arbitrary basis.

Projecting any vector onto a new arbitrary basis as

where each is written in the form

is a system of scalar equations in unknown coordinates

and can be solved using the ordinary Cramers rule for systems of scalar equations, where the step of adding a

basis can be considered as already done. For , the solutions for the scalars are


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For basis vectors ( equations in unknowns), the solution for the th unknown scalar coordinate

generalizes to

the formula for Cramers rule.

Projecting a vector onto an orthogonal basis.

Projections onto arbitrary bases , as solved using Cramers rule as just above, treats projections onto

orthogonal bases as only a special case. Projections onto mutually orthogonalbases can be achieved using the

ordinary projection operation

which is correct only if the are mutually orthogonal.Ifthe bases are constrained to be

mutually perpendicular (orthogonal), then the formula for Cramers rule becomes

where has been written as a sum of vector components parallel and perpendicular to . For any two

perpendicular vectors , , their outer product equals their geometric product. The vector

component must be parallel to the other , therefore its outermorphism is zero. The result is Cramers


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rule reduced to orthogonal projection of vector onto base such that .

In general, the bases are not necessarily mutually orthogonal and the projection to use is

Cramers rule, generalized projection, not the dot product specific to orthogonal projection.

Solving a system of vector equations using SymPy.

The free software SymPy (http://sympy.org), for symbolic mathematics using python, includes a Geometric

Algebra Module and interactive calculator console i sympy. The i sympyconsole can be used to solve systems of

vector equations using the formulas of this article. A simple example of console interaction follows to solve the

system

$i sympy>>> f r om sympy. gal gebra. ga i mpor t *>>> ( e1, e2, e3, e4, e5, e6) = MV. set up( ' e*1| 2| 3| 4| 5| 6' , met r i c=' [ 1, 1, 1, 1, 1, 1] ' )>>> ( v1, v2, v3) = symbol s( ' v1 v2 v3' )>>> ( c1, c2, c3, C) = symbol s( ' c1 c2 c3 C' )>>> ( a1, a2, a3) = symbol s( ' a1 a2 a3' )>>> a1 = 3*e4 + 2*e5 + 9*e6>>> a2 = 4*e4 + 3*e5 + 6*e6>>> a3 = 5*e4 + 7*e5 + 9*e6>>> c1 = 9*e1 + 2*e2 + 3*e3>>> c2 = 6*e1 + 5*e2 + 8*e3>>> c3 = 2*e1 + 4*e2 + 7*e3>>> C = ( c1 e4) + ( c2 e5) + ( c3 e6)>>> v1 = ( C a2 a3) | ( ( - 1)* *( 1-1)*MV. i nv(a1 a2 a3) )>>> v2 = ( a1 C a3) | ( ( - 1)* *( 2-1)*MV. i nv(a1 a2 a3) )>>> v3 = ( a1 a2 C) | ( ( - 1)* *( 3-1)*MV. i nv(a1 a2 a3) )>>> 3*v1 + 4*v2 + 5*v39*e_1 + 2*e_2 + 3*e_3>>> 2*v1 + 3*v2 + 7*v3

6*e_1 + 5*e_2 + 8*e_3>>> 9*v1 + 6*v2 + 9*v32*e_1 + 4*e_2 + 7*e_3

A system of equations is said to be incompatible 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 contrary case the system is

either incompatible or indeterminate, based on the values of the determinants only for 2x2 systems.

For 3x3 or higher systems, the only thing one can say when the coefficient determinant equals zero is: if any of

the "numerator" determinants are nonzero, then the system must be incompatible. However, the converse is false:

having all determinants zero does not imply that the system is indeterminate. A simple example where all

determinants vanish but the system is still incompatible is the 3x3 system x+y+z=1, x+y+z=2, x+y+z=3.


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Matrix

^Cramer, Gabriel (1750). "Introduction l'Analyse des lignes Courbes algbriques" (http://www.europeana.eu/resolve

/record/03486/E71FE3799CEC1F8E2B76962513829D2E36B63015) (in French). Geneva: Europeana. pp. 656659.

Retrieved 2012-05-18.

1.

^MacLaurin, Colin (1748).A Treatise of Algebra, in Three Parts.(http://archive.org/details

/atreatisealgebr03maclgoog).

2.

^Boyer, Carl B. (1968).A History of Mathematics(2nd ed.). Wiley. p. 431.3.

^Katz, Victor (2004).A History of Mathematics(Brief ed.). Pearson Education. pp. 378379.4.

^Hedman, Bruce A. (1999). "An Earlier Date for "Cramer's Rule" " (http://professorhedman.com/Cramers.Rule.pdf).

Historia Mathematica. 4(26) (4): 365368. doi:10.1006/hmat.1999.2247 (http://dx.doi.org

/10.1006%2Fhmat.1999.2247)

5.

^Ken Habgood, Itamar Arel (2012). "A condensation-based application of Cramers rule for solving large-scale linear

systems" (http://web.eecs.utk.edu/~itamar/Papers/JDA2011.pdf).Journal of Discrete Algorithms10: 98109.doi:10.1016/j.jda.2011.06.007 (http://dx.doi.org/10.1016%2Fj.jda.2011.06.007).

6.

^Robinson, Stephen M. (1970). "A Short Proof of Cramer's Rule".Mathematics Magazine43: 9495.7.

Proof of Cramer's Rule (http://planetmath.org/encyclopedia/ProofOfCramersRule.html)

WebApp descriptively solving systems of linear equations with Cramer's Rule (http://sole.ooz.ie/)

Online Calculator of System of linear equations (http://www.elektro-energetika.cz/calculations

/linrov.php?language=english)

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