Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

중앙대학교 건설환경플랜트공학과 교수

김 진 홍

- 2주차 강의 내용 -

7.3 Linear Systems of Equations. Gauss Elimination

Linear System, Coefficient Matrix, Augmented Matrix

 A linear system of m equations in n unknowns is a set of eqs. of the form

(1)

1 1

1

11x a x b

a    _{n n} 

2 2

1

21x a x b

a   _{n n} 

















m n mn

m x a x b

a 1 1   

* ; coefficients^a11,^,amn

are all zero : homogeneous system cf) nonhomogeneous system bm

b₁,,

 Matrix Form of the Linear System (1)

(2) Ax = b

coefficient matrix A =[a_jk] m x n matrix

*

A x b

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

 







 









 

 





 

 









 

n mn

m m

n n

x x x a

a a

a a

a

a a

a

A











1

2 1

2 22

21

1 12

11

,

 Augmented matrixA~

























 

m mn

m m

n n

b a

a a

a a

a

b a

a a

A









2 1

2 22

21

1 1

12 11

~

Ex. 1) Geometric Interpretation, Existence and Uniqueness of Solutions If m  n  2, There are three cases :

(a) Precisely one solution if the lines intersect.

(b) Infinitely many solutions if the lines coincide.

(c) No solutions if the lines are parallel.

2 2 22 1 21

2 2 12 1 11

b x a x a

b x a x a









 







 









b

b



1

and

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

Gauss Elimination and Back Substitution

 If a system is in triangular form, say,

26 13

2 5

2

2 2 1







 x x x

we can solve it by back substitution, and

This gives us the idea of first reducing a general system to triangular form.

22

x x₁ 6.

Ex. 2) Gauss Elimination.

Solve the linear system x₁  x₂  x₃  0

3 0

2

1   

 x x x

90 25

10x₂  x₃  80 10

20x₁ x₂  A~

Augmented Matrix ^{Pivot →}



























80 90 0 0

0 10 20

25 10 0

1 1 1

1 1 1 Eliminate →

30 3

4

2 5

2

2 1

2 1











 x x

x x

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

Step 1. Elimination of x1 Step 2. Elimination of x2

























80 90 0 0

20 30

0

25 10 0

0 0 0

1 1 1



























0 190 90

0

0 0 0

95 0

0

25 10 0

1 1 1

Back Substitution x3^2, x2 ^4, x1 ²

























0 80 90 0

0 0 0

20 30

0

25 10 0

1 1 1

Elementary Row Operations. Row-Equivalent Systems

• Elementary Row Operations for Matrices:

* Interchange of two rows

* Addition of a constant multiple of one row to another row

* Multiplication of a row by a nonzero constant c

Row2 +Row1 Row4-20Row1

Row3-3Row2

3 0

2

1  x  x 

x

90

= 25 + 10x₂ x₃

190

= 95x₃

•We call a linear system S₁ row-equivalent to a linear system S₂ if S₁ can be obtained from S₂ by the above row operations.

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

Ex 3) Solve the linear system 2x₁  6x₂  x₃ 7 1 2 _{2 3}

1 x  x  

x

9 4 7

5x₁ x₂ x₃ A~

Augmented Matrix





















 9

1 7

4 7 5

1 2 1

1 6 2

Step 1. Elimination of Step 2. Elimination of

Back Substitution x3^5, x2^3, x1¹⁰ Row2-2Row1

Row4-5Row1 Row3+1.5Row2

x1 x₂















 





9 7 1

4 7 5

1 6 2

1 2 Row Operation 1















 





14 9

1

1 3 0

3 2 0

1 2 1















  

5 . 27

9 1

5 . 5 0 0

3 2 0

1 2 1

1 2 _{2 3}

1 x x  x

9 3 2x₂  x₃ 

5 . 27 5 . 5 x₃ 

 Theorem 1

Row-Equivalent Systems

Row-Equivalent Systems have the same set of solutions

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

Gauss Elimination: The Three Possible Cases of Systems - unique solution, infinitely many solutions and no solutions

Ex. 4) Gauss Eliminations if Unique Solution Exists

Solve the following linear systems of three equations in three unknowns.

6 4 5 2

3 2 3

1 2

3 2 1

3 2 1

3 2 1





















x x x

x x x

x x x

Step 1 ; Elimination of x1























8 8 3 0

4 4 2 0

1 2 1 1

Step 2 ; Elimination of x2























6 4 5 2

3 2 3 1

1 2 1 1

Row2-Row1

Row3-2Row1 Row3-1.5Row2 



















2 2 0 0

4 4 2 0

1 2 1 1

Thus,

2 2

4 4 2

1 2

3 3 2

3 2 1

















x x x

x x x

1 , 0 ,

1 _{2 1}

3  

x x x

Back Substitution

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

Ex. 5) Gauss Eliminations if Infinitely Many Solutions Exist

Solve the following linear systems of three equations in four unknowns.

























1 . 2 4 . 2 3 . 0 3 . 0 2 . 1

7 . 2 4 . 5 5 . 1 5 . 1 6 . 0

0 . 8 0 . 5 0 . 2 0 . 2 0 . 3

1 . 2 4 . 2 3 . 0 3 . 0 2 . 1

7 . 2 4 . 5 5 . 1 5 . 1 6 . 0

0 . 8 0 . 5 0 . 2 0 . 2 0 . 3

4 3

2 1

4 3

2 1

4 3

2 1

























x x

x x

x x

x x

x x

x x

Step 1 ; Elimination ofx1



























1 . 1 4 . 4 1 . 1 1 . 1 0

1 . 1 4 . 4 1 . 1 1 . 1 0

0 . 8 0 . 5 0 . 2 0 . 2 0 . 3

Row2-0.2Row1 Row3-0.4Row1

Step 2 ; Elimination ofx2





















0 0 0 0 0

1 . 1 4 . 4 1 . 1 1 . 1 0

0 . 8 0 . 5 0 . 2 0 . 2 0 . 3

Row3+Row2

0 0

1 . 1 4 . 4 1 . 1 1 . 1

0 . 8 0 . 5 0 . 2 0 . 2 0 . 3

4 3

2

4 3

2 1

















x x

x

x x

x Thus, x

Back Substitution x2 1x34x4, x1 2x4

← infinitely many solutions since and remain arbitraryx₃ x₄

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

Ex. 6) Gauss Eliminations if no Solution Exists

Solve the following linear systems of three equations in three unknowns.

6 4 2

6

0 2

3 2

3

3 2

1

3 2

1

3 2

1



















x x

x

x x

x

x x

x

















6 4 2 6

0 1 1 2

3 1 2 3

Step 1 ; Elimination of x1























0 2 2 0

2 3 / 1 3 / 1 0

3 1 2 3

Row2-(2/3)Row1 Row3-2Row1

Step 2 ; Elimination of x2





















12 0 0 0

2 3 / 1 3 / 1 0

3 1 2 3

Row3-6Row2

12 0

2 )

3 / 1 ( ) 3 / 1 (

3 2

3

3 2

3 2

1

















x x

x x

x

The false statement shows that the system has no solution. 0 12 Thus,

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

7.6 Second and Third Order Determinants

- Determinants of second order

(1) _{11 22 12 21}

22 21

12

det 11 a a a a

a a

a A a

D    

- Cramer's rule for solving linear systems of two equations

(2) (a)

2 2 22 1 21

1 2 12 1 11

b x a x a

b x a x a







 is, (b)

(3) 1 ^1,

D x  D

D

x₂  D² with D≠0

where,

22 21

12 11

a a

a D  a

22 2

12 1

1 b a

a D  b

2 21

1 11

2 a b

b D  a

21 12 22

11a a a

a 

 b₁a₂₂ a₁₂b₂  a₁₁b₂ b₁a₂₁

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

Ex. 1) Cramer's Rule for Two Equations

8 5

2

12 3

4

2 1

2 1









 x x

x x

14 4 56

5 2

3 4

8 2

12 4 ,

14 6 84

5 2

3 4

5 8

3 12

2

1     x      

x

Third-Order Determinants

 Determinants of third order

(4)

23 22

13 12 31 33 32

13 12 21 33 32

23 22 11 33 32 31

23 22 21

13 12 11

a a

a a a

a a

a a a

a a

a a a

a a a

a a a

a a a

D   

* signs ; + - +

* entry in the first column of D times its minor

* minor ; the second-order determinant obtained from D by deleting the row and column of that entry

(4*) D













Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

Cramer's Rule for Linear Systems of Three Equations (5)

3 3 33 2 32 1 31

2 3 23 2 22 1 21

1 3 13 2 12 1 11

b x a x a x a

b x a x a x a

b x a x a x a



















(6) D

x D D

x D D

x₁ D¹, ₂  ², ₃ ³

where,

3 32 31

2 22 21

1 12 11 3 33 3 31

23 2 21

13 1 11 2 33 32 3

23 22 2

13 12 1

1 , ,

b a a

b a a

b a a D a b a

a b a

a b a D a

a b

a a b

a a b

D   

Ex. 2) Cramer's Rule to Solve Equations

1 2 4 5

3 3

7 2

3

3 2 1

3 2 1

3 2 1



















x x x

x x x

x x x

,

33 32 31

23 22 21

13 12 11

a a a

a a a

a a a D

, 13 2 4 5

3 1 1

1 2 3







D 39

2 4 1

3 1 3

1 2 7

1 





 D

, 78 2 1 5

3 3 1

1 7 3

2 





D 52

1 4 5

3 1 1

7 2 3

3   

4 D ,

6 ,

3 ₂ ² ₃ ³

1

1      

D x D D

x D D

x D

Chap. 7 Linear Algebra : Matrices, Vectors, Determinants

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