2019학년도 1학기

반도체 공학과 공학수학1

주교재: Erwin Kreyszig, Engineering Mathematics, 10th Edition

부교재: 이상구 외 4인, 최신공학수학 I,  1st Edition

강의시간:  공학수학1, (화09:00-10:15)   (목10:30-11:45)

담당교수:  김응기 박사

Week 13

13주차

7.6: For Reference: Second and Third-Order Determinants

7.7: Determinants. Cramer’s Rule

1.4 행렬식과 여인자 전개

7.8: Inverse of Matrix

1.5 역행렬과 크래머 법칙

7.6 For Reference: Second-order and Third-order Determinants

Second-Order Determinants

A determinant of second-order is denoted and defined by

(1)

Cramer's rule for solving linear systems of two equations

(2)

(3)

with as in (1), provided .

The value appears for homogeneous systems with nontrivial solutions.

Example

Compute the determinant of the following matrix.

Solution

Sage Coding

 A = matrix([[3, 1], [4, -2]]) print A.det()

Evaluate

-10

Example 1  Cramer's Rule for Two Equations

.

Then .

Sage Coding

 A = matrix([[4, 3], [2, 5]]) b = vector([12, -8]) A1 = column_matrix([b, A.column(1)]) A2 = column_matrix([A.column(0), b]) print A.det() print A1.det() print A2.det() print "x =",  A1.det()/A.det() print "y =",  A2.det()/A.det()

Evaluate

14

84

-56

x = 6

y = -4

Third-Order Determinants

Determinant of Third-order is

(4)

Cramer's rule for solving linear systems of third equations

(5)

(6)

with the determinant of the system given by (4) and

.

Example

Compute the determinant of the following matrix.

Solution

.

Sage Coding

 B = matrix([[1, 2, 3], [-4, 5, 6], [7, -8, 9]]) print B.det()                   # compute the determinant

Evaluate

240

Example

Find the determinant of a matrix , where

Solution

Sage Coding

 A = matrix([[0, 1, 5], [3, -6, 9], [2, 6, 1]]) print A.det()

Evaluate

165

Example

Solve the following system of linear equations by Cramer's rule.

Solution

Let be the coefficient matrix. Then

, and hence

.

■                      Sage Coding

 A = matrix([[-2, 3, -1], [1, 2, -1], [-2, -1, 1]]) b = vector([1, 4, -3]) A1 = column_matrix([b, A.column(1), A.column(2)]) A2 = column_matrix([A.column(0), b, A.column(2)]) A3 = column_matrix([A.column(0), A.column(1), b]) print A.det() print A1.det() print A2.det() print A3.det() print "x =",  A1.det()/A.det() print "y =",  A2.det()/A.det() print "z =",  A3.det()/A.det()

Evaluate

-2

-4

-6

-8

x = 2

y = 3

z = 4

7.7. Determinant. Cramer's Rule

Determinant of order is a scalar associated with an matrix , is

(1)          .

Minors and Cofactors

Consider an square matrix . Let denote the square submatrix of obtained by deleting its th row and th column. The determinant is call the minor of the element of , and we define the cofactor of , denoted by .

Cofactor matrix

The matrix obtained from by replacing the th row of by the th row.

The matrix obtained from by replacing the th column of by the th column.

Example 1  Minors and Cofactors a Third-Order Determinant

Minors matrix and Cofactors matrix.

Solution

Example 2  Expansions of a Third-Order Determinant

Sage Coding

 D = matrix(3, 3, [1, 3, 0, 2, 6, 4, -1, 0, 2]) D.det()

Evaluate

-12

Example 3  Determinant of a Triangular Matrix

Sage Coding

 D = matrix(3, 3, [-3, 0, 0, 6, 4, 0, -1, 2, 5]) D.det()

Evaluate

-60

General Properties of Determinants

Theorem 1 Behavior of an -Order Determinant under Elementary Row Operations

(a) Interchange of two rows multiplies the value of the determinant by .

(b) Addition of a multiple of a row to another row does not alter the value of the determinant.

(c) Multiplication of a row by a nonzero constant multiplies the value of the determinant by . (This holds also when , but no longer gives an elementary row operation.)

Example 4  Evaluation of Determinants by Reduction to triangular Form

Find .

Solution

.

Sage Coding

 A = matrix(4, 4, [2, 0, -4, 6, 4, 5, 1, 0, 0, 2, 6, -1, -3, 8, 9, 1]) A.det()

Evaluate

1134

Theorem 2  Further Properties of an Order Determinants

(a) Interchange of two rows multiplies the value of the determinant by .

(b) Addition of a multiple of a row to another row does not alter the value of the determinant.

(c) Multiplication of a row by a nonzero constant multiplies the value of the determinant by . (This holds also when , but no longer gives an elementary row operation.)

(d) Transposition leaves the value of a determinant unaltered.

(e) A zero row or column renders the value of a determinant zero.

(f) Proportional rows or column render the value of a determinant zero.

In particular, a determinant with two identical rows or columns has the value zero.

Theorem 3  Rank in terms of Determinants

Consider an matrix :

(1) has rank     has an submatrix with a nonzero determinant.

(2) The determinant of any square submatrix with more than rows, contained in has a value equal to zero.

Furthermore, if  , we have :

(3) An square matrix has rank       .

Cramer’s Rule

If a linear system of equations in the same number of unknowns , , ,

has a non-zero coefficient determinants , the system has precisely one solution. This solution is given by the formulas

,           (Cramer’s rule)

where is the determinant obtained from by replacing in the th column by the column with the entries , , , .

(b) Hence if the system is homogeneous and , it has only the trivial solution , , , . If , the homogeneous system also has nontrivial solutions.

7.8 Inverse of a Matrix. Gauss-Jordan Elimination

The inverse of an matrix is denoted by such that

where is the unit matrix.

If has an inverse then is a nonsingular matrix.

If has an no inverse then is a singular matrix.

If has an inverse, the inverse is unique.

If both and are inverse of , then and . Show that .

Theorem 1  Existence of the Inverse

The inverse of matrix exists if and only if , thus if and only if . Hence is nonsingular if , and is singular if .

Determination of the Inverse by the Gauss-Jordan Elimination

Example 1  Inverse of a Matrix. Gauss-Jordan Elimination

Determine the inverse of .

Solution

The last columns constitute .

. Similarly .

Sage Coding

 A = matrix([[-1, 1, 2], [3, -1, 1], [-1, 3, 4]]) I = identity_matrix(3) Aug = A.augment(I).rref() # RREF of the augmented matrix [A : I] print Aug print print Aug.submatrix(0, 3, 3, 3)  # 역행렬 print print A.inverse() # 내부 명령어(역행렬)

Evaluate

[     1      0      0  -7/10    1/5   3/10]

[     0      1      0 -13/10   -1/5   7/10]

[     0      0      1    4/5    1/5   -1/5]

[ -7/10    1/5   3/10]

[-13/10   -1/5   7/10]

[   4/5    1/5   -1/5]

[ -7/10    1/5   3/10]

[-13/10   -1/5   7/10]

[   4/5    1/5   -1/5]

Useful Formulas for Inverses

Theorem 2  Inverses of a Matrix

The inverse of a nonsingular matrix is given by

where is the cofactor of in .

In particular, the inverse of

is .

Example 2  Inverses of a Matrix

.

Sage Coding

 A=matrix(2, 2, [3, 1, 2, 4]) print "A^(-1)=" print A.inverse()

Evaluate

A^(-1)=

[  2/5 -1/10]

[ -1/5  3/10]

Example 3  Inverses of a Matrix

Find the inverse of

Solution

Sage Coding

 A = matrix(3, 3, [-1, 1, 2, 3, -1, 1, -1, 3, 4]) detA = A.det() print 1/detA*A.adjoint()

Evaluate

[ -7/10    1/5   3/10]

[-13/10   -1/5   7/10]

[   4/5    1/5   -1/5]

Diagonal matrices , when , have an inverse      all . Then is diagonal, too with entries , , , .

Example 4  Inverse of a Diagonal Matrix

Let .

Solution

Then the inverse is .

Products can be inverted by taking the inverse of each factor and multiplying these inverses in reverse order,

.

Hence for more than two factors,

.

Unusual Properties of Matrix Multiplication. Cancellation Laws

[1] Matrix multiplication is not commutative, that is, in general we have

.

[2] does not generally imply or . For example,

.

[3] does not generally imply (even when ).

Theorem 3  Cancellation Laws

Let be matrices, then

(a) If and then .

(b) If , then implies . hence if , but as well as , then and .

(c) If is singular, so are and .

Determinants of Matrix Products

Theorem 4  Determinant of a Product Matrices

For any matrices and ,

.

Theorem  Determinant of a sum Matrices

For any matrices and ,

.

and are matrix.

(1)

(2)

Solution

,

,

(1)

(2)

Sage Coding

 A = matrix(2, 2, [6, 1, 3, 2]) B = matrix(2, 2, [4, 3, 1, 2]) print A.det() print B.det() print (A.det())*(B.det()) print (A.det()) + (B.det()) print bool(((A*B).det() ) ==  (A.det())*(B.det())) print bool((A.det())*(B.det()) ==  (A.det()) + (B.det()))

Evaluate

9

5

45

14

True

False

[한빛 아카데미] Engineering Mathematics with Sage:

[저자] 이상 구, 김영 록, 박준 현, 김응 기, 이재 화

Contents

A. 공학수학 1 – 선형대수, 상미분방정식+ Lab

Chapter 01 벡터와 선형대수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-1.html

Chapter 02 미분방정식의 이해 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-2.html

Chapter 03 1계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-3.html

Chapter 04 2계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-4.html

Chapter 05 고계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-5.html

Chapter 06 연립미분방정식, 비선형미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-6.html

Chapter 07 상미분방정식의 급수해법, 특수함수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-7.html

Chapter 08 라플라스 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-8.html

B. 공학수학 2 - 벡터미적분, 복소해석 + Lab

Chapter 09 벡터미분, 기울기, 발산, 회전 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-9.html

Chapter 10 벡터적분, 적분정리 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-10.html

Chapter 11 푸리에 급수, 적분 및 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-11.html

Chapter 12 편미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-12.html

Chapter 13 복소수와 복소함수, 복소미분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-13.html

Chapter 14 복소적분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-14.html

Chapter 15 급수, 유수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-15.html

Chapter 16 등각사상 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-16.html

Made by Prof. Sang-Gu LEE  sglee at skku.edu

http://matrix.skku.ac.kr/sglee/   with Dr. Jae Hwa LEE