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# Systems of linear algebraic equations

## 1.

Systems of linear algebraic equations.Сramer rule.

Consider a system of m linear equations with n variables

x1, x2, x3, …, xn:

a11 x1 a12 x 2 ... a1n x n b1 ,

a x a x ... a x b ,

21 1

22 2

2n n

2

..............................................

a m1 x1 a m 2 x 2 ... a mn x n bm .

The coefficients aik (i = 1, 2, …, m; k = 1, 2, …, n) of

unknowns are enumerated by two indices: the first index i

indicates the number of the equation, and the second j –

the number of the variable.

## 2.

Solving a system of linear equations by means ofdeterminants

(Cramer rule)

Consider a system of two equations with two unknowns

(*):

a11 x a12 y b1 ,

a21 x a22 y b2 .

a11

a12

a

a

The determinant

is called the basic

determinant of the system of equations (*), and the

determinants b1 a12 a b

21

22

11

x

b2

a22

y

1

a21 b2

and are called the auxiliary determinants of the system.

## 3.

The following cases are possible:1) 0.

In this case each equation of the system (*) has a

unique decision: x , Cramer formulas.

x

y y .

2) = 0, but at least one of х, у is not equal to zero.

In this case there is at least one equation of the system

which has no decision. Consequently the original

system (*) has no decision, i.e. is non-compatible.

3) = х = у = 0.

In this case each equation of the system has infinitely

many decisions. Consequently the original system (*)

is compatible and indeterminate.

## 4.

Example 1. Solve the system2 x 3 y 7,

3x 4 y 2.

Solution: 1) Compute the basic determinant of the system

2 3

3

4

8 9 17.

2) Compute the auxiliary determinants of the system:

x

7 3

2

28 6 34; y

2 7

4 21 17.

3 2

= 17 0 x x 34 2; y y 17 1.

17

17

4

2 ( 2) 3 1 7

3 ( 2) 4 1 2

## 5.

a11 x a12 y a13 z b1a 21 x a 22 y a 23 z b2

a x a y a z b

32

33

3

31

the Cramer formulas are deduced analogously and have the

following form:

y

x

x

, y

,

z

z

a11

a12

a13

b1

a12

a13

a 21

a 22

a 23 0, x b2

a 22

a31

a32

a33

a32

b3

a11

a11

a12

b1

a 23 , y a 21 b2

a 23 , z a 21

a 22

b2 .

a33

a33

a32

b3

a31

b1

b3

a13

a31

## 6.

Example 2. Solve the system of equations:2 x 3 y z 5,

x 2 y 3 z 4,

3 x y 2 z 1.

Solution. Compute the basic and auxiliary determinants of the

system:

2

3

1

3 8 27 1 6 6 6 18 0 x 4

2

3

1

2

5 3

1

1

2

5

2

1

1

3 20 9 4 2 24 1 18,

2

1

y 1 4 3 16 45 1 12 10 6 36,

3

2

z 1

3

1

2

3 5

2

1

4 4 36 5 30 3 8 54.

1

x

y 36

x 18

54

1; y

2; z z

3.

18

18

18