mat 1275: introduction to mathematical analysis dr. a. rozenblyum

MAT 1275: Introduction to

Mathematical AnalysisDr. A. Rozenblyum

IV. Systems of Equations

D. Determinants and Cramer’s Rule

Determinants and Cramer’s Rule

In sections IV.A – IV.C, we considered methods of solving systems of linear equations based on some specific operations upon the equations of given systems. Here we consider formulas that allow to calculate solutions explicitly without manipulations with equations, but by direct substitution of coefficients of equations into these formulas. Such formulas are called the Cramer’s rule named after Gabriel Cramer (1704 – 1752), a Swiss mathematician. Cramer’s rule is not efficient for systems with many equations, and is not used in practical calculations. However, it is easy to use for systems with two and three equations that we consider here. Also, is has a theoretical importance.

Case of the system with two equations

Let’s derive Cramer’s rule for the system

feydx

cbyax.

In section IV.B we considered some specific examples of using the elimination method. Let’s apply this method in general form to the above system. To find x, we eliminate y by multiplying the first equation by e, the second equation by –b, and adding the resulting equations:

feydx

cbyax

bfbeybdx

cebeyaex aex – bdx = ce – bf,

(ae – bd)x = ce – bf, bdae

fbcex

.

e

– b

In similar way we can find y by eliminating x:

feydx

cbyax

afaeyadx

cdbdyadx – bdy + aey = –cd + af,

(ae – bd)y = af – cd, bdae

cdafy

.

– d

a


We come up to the following general formulas for the solutions of the system of two linear equations with two variables:

bdae

fbcex

, bdae

cdafy

.

Observe these formulas. Notice that the denominators of both fractions are the same, and numerators look similar to denominators. Cramer’s rule represents these formulas in terms of a special number that is called the determinant. For the system of two equations, determinant is defined by four numbers, say k, l, m, and n. Here is the notation and the definition of the determinant:

mlknnm

lk .

We call it 22 determinant. As you can see, to calculate it, we take the product along the main diagonal (from left top corner to right bottom corner) minus the product along the minor diagonal (from left bottom corner to right top corner). In this determinant, numbers k and m form the first column, and numbers l and n form the second column.

If you return back to the formulas for the solutions x and y, you may notice that their numerators and denominators can be written in terms of determinants. We come up to the following Cramer’s rule.


Solution of the system

feydx

cbyax

includes these steps:

1) Calculate the following determinant D which is called the determinant of the system:

bdaeed

baD .

Notice that “free” coefficients c and f from the right side of the system are not used in the determinant D. It consists of the coefficients for x and y only.

2) Calculate another two determinants, xD and yD:

bfceef

bcDx , cdaf

fd

caDy .

Notice that determinant xD is obtained from D by replacing its first column with the column of “free” coefficients c and f from the right side of the system. Similar, determinant yD is obtained from D by replacing its second column with

the column of “free” coefficients.

3) Calculate the solutions of the system by the formulas

D

Dxx,

D

Dyy.

Note: As you see, these formulas contain determinant D in the denominator. Therefore, these formulas makes sense only if 0D. If D = 0, then the system does not have unique solution. Instead, it either does not have solutions at all, or it has infinite number of solutions. To detect which case we have, we should check xD (or yD) for zero. If

0xD, then there are no solutions. If 0xD, then the system has infinite number of

solutions. (It can be shown that if D = 0, both xD and yD are equal or not equal to zero

simultaneously).


Example 1. Solve the following system using the Cramer’s rule.

735

427

yx

yx.

Solution. 1) Calculate the determinant D of the system:

311021)2(53735

27

D .

2 ) C a lc u la t e t h e d e t e rm in a n t s xD a n d yD :

261412)2(73437

24

xD ,

.292049457775

47yD

3) Write the solutions of the system

31

26

D

Dx x ,

31

29

D

Dy y

.

Final answer: 31

26x ,

31

29y , or, as a solution set,

31

29,31

26.


Case of the system with three equations

Consider the Cramer’s rule for the system

3333

2222

1111

dzcybxa

dzcybxa

dzcybxa

.

Similar to systems with two equations, the solutions of this system can also be represented in terms of determinants as ratios of determinants xD, yD, and zD

corresponding to variables x, y, and z, to the common determinant D of the system. Let’s describe how to define these determinants.


We will not derive here corresponding formulas, and just provide the final result. The determinant D of the above system is denoted by

333

222

111

cba

cba

cba

D .

This is a 33 determinant. There are several different methods how to calculate it. We consider two methods: direct calculation and expansion-by-minors.

Direct calculation method. Here is the formula

132321321 cbaacbcbaD

132321321 abccababc .

This formula looks rather complicated and seems difficult to memorize. A possible way to memorize it is this. Notice that the formula contains six terms: three with the plus sign, and another three with the minus sign. Three terms with the plus sign correspond to the main diagonal 321 ,, cba of the determinant D: one of these terms is 321 cba , which is the

product along main diagonal, and two others, 321 acb and 132 cba , are products

corresponding to small diagonals 21,cb and 32,ba which are parallel to the main diagonal.

Similar structure is for three terms with the minus sign: one of them, 321abc, is the product

along the minor diagonal 321,,abc , and two others, 321cab and 132abc, are products that

correspond to small diagonals 21ab and 32bc which are parallel to the minor diagonal.


Here is another way to memorize the above formula. Let’s extend (double) the determinant D to the following table:

333333

222222

111111

cbacba

cbacba

cbacba

.

Then to get three terms of the determinant with the plus sign, calculate products along the main diagonal 321,,cba , and two parallel diagonals 321,,acb and 321,,bac .

To get three terms with the minus sign, calculate products along the minor diagonal

123,,cba , and two parallel diagonals 123,,acb and 123,,bac .

Note. The last column of the above table is not used, so it is not necessary to write it.

Example 2. Calculate the following determinant by direct calculation method

302

423

265

D .

Solution. Construct the extended matrix (without last column)

02302

23423

65265

W e h a v e

03)2(2)4()6(325 D

1405484830)6(335)4(0)2(22 .


Expansion-by-Minors method. First of all, let’s define what is the minor. We can define a minor for any element of the determinant D. This is a 22 determinant that is obtained from the determinant D by erasing a row and a column in which given element is located. Let’s denote a minor that corresponds to some element by the same but capital letter as the element. For example, for element 1a the corresponding minor is

33

221 cb

cbA , and for element 2b, the minor is

33

112 ca

caB .

To describe expansion-by-minors method, we first “assign a sign” (plus or minus) to each element of the determinant D. The following rule is used: for a given element calculate the sum of its row number and column number. If this sum is even, assign plus, if the sum is odd, assign minus. For example, element 1a is located in the first row and first

column, the sum is 2 (1+ 1). This number is even, so we assign plus sign to 1a. Another

example: element 2c is located in the second row and third column, the sum is 5 (2 + 3).

This number is odd, so we assign minus sign to2c. Here is the complete picture of signs for all element of the determinant D:

.

Notice that when you move along any row or any column, the signs alternate.


Expansion-by-minors method works with any row and any column of the determinant D. Let’s choose, for example, 2nd row 222 ,, cba . We multiply each element of this row by its minor, assign to this product corresponding sign and add all of them. As a result, we obtain the determinant D:

222222 CcBbAaD .

We call this formula the expansion of the determinant D over the second row. As another example, consider the expansion of the determinant D over the third column:

332211 CcCcCcD . This method works especially effective when determinant D contains many zeros. In this case, select a row or a column with biggest number of zeros.

Example 3. Calculate the determinant D from example 2 by expansion-by-minors method

302

423

265

D .

S o l u t i o n . T h i s d e t e r m i n a n t c o n t a i n s z e r o i n 3 r d r o w a n d 2 n d c o l u m n . T h e r e f o r e , i t i s a g o o d i d e a t o e x p a n d i t o v e r 3 r d r o w o r 2 n d c o l u m n . W e c h o o s e 2 n d c o l u m n :

43

250

32

252

32

43)6(

D

= 140192176)2(2352)4(2336 .


Now, we are ready to describe the Cramer’s rule for the system

3333

2222

1111

dzcybxa

dzcybxa

dzcybxa

.

1) Calculate the determinant D of this system:

333

222

111

cba

cba

cba

D .

2) Calculate three other determinants, xD, yD and zD that correspond to variables

x, y, and z. These determinants are constructed by replacing corresponding columns of determinant D with the column from the right side of the system:

333

222

111

cbd

cbd

cbd

Dx ,

333

222

111

cda

cda

cda

Dy ,

333

222

111

dba

dba

dba

D .

3) Calculate the solutions of the system by the formulas

D

Dxx,

D

Dyy,

D

Dzz.

Note: Similar to the case of the system of two variable, there are no solution or there is infinite number of solutions, if D = 0.


Example 4. Solve the following system using the Cramer’s rule.

532

8423

7265

zx

zyx

zyx

Solution.

1) The determinant D of the system is

302

423

265

D .

This is exactly the same determinant as in examples 2 and 3, so D = 140.

2) Calculate the determinants, xD, yD and zD.

305

428

267

xD ,

352

483

275

yD ,

502

823

765

zD .

W e e x p a n d d e t e r m i n a n t xD o v e r t h e 3 r d r o w :

28

673

42

265

xD

= 38)6()8(273)2(2)4()6(5 ,


W e e x p a n d d e t e r m i n a n t yD o v e r t h e 1 s t c o l u m n :

48

272

35

273

35

485

yD

= 201)2()8()4(72)2(5373)4(53)8(5 W e e x p a n d d e t e r m i n a n t zD o v e r t h e 2 n d c o l u m n :

52

752

52

83)6(

zD

= 20872552)8(2536 .

3) Calculate the solutions x, y, and z of the system

70

19

140

38

D

Dx x ,

140

201

D

Dy y

, 35

52

140

208

D

Dz z .

Final answer: 35

52,

140

201,

70

19 zyx , or, as a solution set,

35

52,

140

201,

70

19.

End of the Topic

mat 1275: introduction to mathematical analysis dr. a. rozenblyum

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