Cramer's Rule Gabriel Cramer was a Swiss mathematician (1704-1752)

Reference from: http://www.mathcentre.ac.uk : Fundamentals Methods of Mathematical Economics 4th Edition (Page 103-107)

Introduction Cramer’s Rule is a method for solving linear

simultaneous equations. It makes use of determinants and so a knowledge of these is necessary before proceeding.

Cramer’s Rule relies on determinants

Coefficient Matrices You can use determinants to solve a system of

linear equations. You use the coefficient matrix of the linear

system. Linear System Coeff Matrix

ax+by=ecx+dy=f

dc

ba

Cramer’s Rule for 2x2 System Let A be the coefficient matrix Linear System Coeff Matrix

ax+by=ecx+dy=f

If detA 0, then the system has exactly one solution:

and

A

df

be

xdet

A

fc

ea

ydet

dc

ba= ad – bc

Key Points The denominator consists of the coefficients

of variables (x in the first column, and y in the second column).

The numerator is the same as the denominator, with the constants replacing the coefficients of the variable for which you are solving.

Example - Applying Cramer’s Ruleon a System of Two Equations

Solve the system: 8x+5y= 2 2x-4y= -10

The coefficient matrix is: and

So:

and

42

5842)10()32(

42

58

42

410

52

x42

102

28

y

142

42

42

)50(8

42

410

52

x

242

84

42

480

42

102

28

y

Solution: (-1,2)

Applying Cramer’s Ruleon a System of Two Equations

D

Dy

D

Dx

fc

eaD

df

beD

dc

baD

fdycx

ebyax

yx

y

x

1453

1632

yx

yx

19910)3)(3()5)(2(53

32

D

384280)14)(3()5)(16(514

316

xD

764828)16)(3()14)(2(143

162

yD

419

762

19

38

D

Dy

D

Dx yx

Evaluating a 3x3 Determinant(expanding along the top row)

Expanding by Minors (little 2x2 determinants)

33

22

1

33

22

1

33

22

1

333

222

111

ba

bac

ca

cab

cb

cba

cba

cba

cba

23896

)4)(2()3)(3()6)(1(

21

02)2(

31

32)3(

32

30)1(

321

302

231

Using Cramer’s Rule to Solve a System of Three Equations

Consider the following set of linear equations

11 1 12 2 13 3 1

21 1 22 2 23 3 2

31 1 32 2 33 3 3

a x a x a x b

a x a x a x b

a x a x a x b

Using Cramer’s Rule to Solve a System of Three Equations

The system of equations above can be written in a matrix form as:

11 12 13 1 1

21 22 23 2 2

31 32 33 3 3

a a a x b

a a a x b

a a a x b

Using Cramer’s Rule to Solve a System of Three Equations

Define

11 12 13

21 22 23

31 32 33

1 1

2 2

3 3

a a a

A a a a

a a a

x b

x x and B b

x b

31 21 2 3

If 0, then the system has a unique solution

as shown below (Cramer's Rule).

, ,

D

DD Dx x x

D D D

Using Cramer’s Rule to Solve a System of Three Equationswhere

11 12 13 1 12 13

12 22 23 1 2 22 23

13 32 33 3 32 33

11 1 13 11 12 1

2 12 2 23 3 12 22 2

13 3 33 13 32 3

a a a b a a

D a a a D b a a

a a a b a a

a b a a a b

D a b a D a a b

a b a a a b

Example 1Consider the following equations:

1 2 3

1 2 3

1 2 3

2 4 5 36

3 5 7 7

5 3 8 31

where

2 4 5

3 5 7

5 3 8

x x x

x x x

x x x

A x B

A

Example 1

1

2

3

36

7

31

x

x x and B

x

2 4 5

3 5 7 336

5 3 8

D

1

36 4 5

7 5 7 672

31 3 8

D

Example 1

2

2 36 5

3 7 7 1008

5 31 8

D

3

2 4 36

3 5 7 1344

5 3 31

D

11

22

33

6722

3361008

33361344

4336

Dx

DD

xDD

xD

Cramer’s Rule - 3 x 3 Consider the 3 equation system below with variables x, y and z:

a1x b1y c1z C1

a2x b2y c2z C2

a3x b3y c3z C3

Cramer’s Rule - 3 x 3 The formulae for the values of x, y and z are

shown below. Notice that all three have the same denominator.

x

C1 b1 c1

C2 b2 c2

C3 b3 c3

a1 b1 c1

a2 b2 c2

a3 b3 c3

y

a1 C1 c1

a2 C2 c2

a3 C3 c3

a1 b1 c1

a2 b2 c2

a3 b3 c3

z

a1 b1 C1

a2 b2 C2

a3 b3 C3

a1 b1 c1

a2 b2 c2

a3 b3 c3

Example 1 Solve the system : 3x - 2y + z = 9 x + 2y - 2z = -5

x + y - 4z = -2

x

9 2 1

5 2 2

2 1 4

3 2 1

1 2 2

1 1 4

23

231 y

3 9 1

1 5 2

1 2 4

3 2 1

1 2 2

1 1 4

69

23 3

Example 1

z

3 2 9

1 2 5

1 1 2

3 2 1

1 2 2

1 1 4

0

230

The solution is

(1, -3, 0)

Cramer’s Rule Not all systems have a definite solution. If the

determinant of the coefficient matrix is zero, a solution cannot be found using Cramer’s Rule because of division by zero.

When the solution cannot be determined, one of two conditions exists: The planes graphed by each equation are parallel

and there are no solutions. The three planes share one line (like three pages of

a book share the same spine) or represent the same plane, in which case there are infinite solutions.