1969-04-17 — Page 22

貫二第張六第日一初月三年西巴瑟夏

WAH KIU YAT

報日僑華

日七十月四年九六九一公年八十五國民華中育教僑華

FELS PES #SES JELAS.

The determinant of a square matrix à is denoted by Al

僑

Example

A-(-), then the determinant of

6 12 18

0. for in the first and second rows

we have

CX文中學會尺試題預習

現代數學科 (十四):

MODERN MATHEMATION (24)

Solutions of the problems of last weeks

The determinant or a square matrix (a) o1" 1 x 1 order is the number "a" itself. That is,

The determinant of a 2 x 2 matrix”

is the number ai

That 19,

18. 2.

Property 5 If air the elements in ons row or one column of a determinant is multiplied by a constan

, then the value of the determine is multiplied by that bonstant k.

Example:

1104 3

52 61

58

1. Given

and BA

Solutions

2

Find

The determinant of 3x3 matrix.

11 12 13

21 22

223

( : ) ( ) ( )

01

BA

3. Express as one satrix

Solutions

( ) ( )) (0.3+(-1))

find

(

the number

*32

[*31

33

above.

That 18

the number expressad

Strictly speaking, a matrix is a rectangular arrangement of numbers whereas a determinant is a number referring to the matrix, Notice that a determinant is in the same order as that of the matrix that gave rise to it. Definition of a determinant will be given in a more general way after we have learned the term "cofactor of a matrix. However there is a more convenient device to express the 3x3 determinants. Let

12 131

thea

21 22 23

31 32 33

Property 6

The value of a determinant remains. unchanged if a multiple of the elements in any row or column be added to the corresponding elements of another row or clumn.

Exampler

2 14 3-16

2

151

The elements of the last column in the right.

hand side are obtained by adding to then the umbers of the first column multiplied by -2.

17. Minor and cofactor

Definition The minor of an element zij of a determinant is formed by the elements of the determinant which are left when the elements the row and column containing alj have been deleted. The cofactor of any element mij is the minor of that element multiplied by (-1)1" denoted by

In the last section we expanded

∙11

22 23

32 33

13

31 32

93 X 13

Solution:

lis

o)

4. IFA-(-1) and D. (19).

BA

Solution

(AB).

10

find (AB) and

[GD] (1.: : :) - ( ) - ( ;)

()()()(-)-(+))

(AB)

B

In doing so, the 3x3 determinant is considered as the sum of three 2 x 2 determinants, the factor in front of each of the 212 determinant being the element of the first row. Notice that correspond ang to each element in the first row we form a related subdeterminant by orossing out the row and column in which the element stands. The determinant may be expanded by the elements of any other row or column with the understanding that the element aij in front of each 2x2 subdeterminant is multiplied by (-1)+). Thus Tal can be expanded as

(a) = (-2)2+*,,, | *22 #23] + (-2)2+1=21

Examples i

13

terms of cofactor, we can expand it as follOWS.

912 X 12

properties of determinante involving cofactors are stated as below.

Property 6 The value of a determinant 18 equal to the sum of the products formed by each element in a row or column, multiplied by its cofactor, Thus, the determinant above can be expressed below.

TAL 12 12

*22 x 22 * *32 × 32

#320 32

#33

and so on.

Property 7 If each element in any row or 001 UMUL of a determinant is aultiplied by the cofactor of the corresponding element in some other row or column, then the results added, give zaro,

Since property 7 is the property upon whion vae inverse of a square matriz relies, we shall give a proof of the property.

Proofs Get A -

11 12 13 and B

21 22 23

31 32 33

11-12-13)

Then 70, because it has 2 identical rows.

But B can be

2223*

Red expanded as a

Comparing the

the last TOW

elements 1214 21

cofactors of the two determinante,

the cofactors of the elements of the last row in

A are the same as the cofactors of the last row in B. Thus, d.

dan be and a

23 replaced by O

and ∞, respectively.

33

That

21/0221

32

31

Let I be the identity, matrix of the order 3 x 3. Find 1′′, 13 and 14. What is your conclusion: shrnt the resul, to...”

Solution

2- (610) (020)-(610)

()()()(0)

1:0

Similarly, 1*

(2)

[C - C

14. Properties of determinants

We shall state the properties or leterminants without proofs.:

Property 1 If the rows of a determinant be come columns and columne bacoma rows, the value of the determinant remains unchanged, Example

is sero. Hence it is provad.

16. Marked examplan

(1) Evaluates Notice that in the solution R denotes row and "O denoten. 100lumas,

brato

or two

(a+b+c)

(a+bro ia

the common factor in

We conclude that any positive integral powers of a matrix identity is the identity itself.

Matrices (continued)

Determinants

Up to now we have considered addition, subtraction, and multiplication of matrices, but not division. In order to construct a Theory of matrix division of matrix inversion we need to review the meaning of the inverse of an operation. We know the additive inverse of a number neispal number such that n+m. Without doubt, a is

n. Similarly the multiplicative inverse of a number p is another number q such that px q. 1 Obviously, gie 1. In an analogous manner a matrix B is the P inverse of A if AxBI, where I is the identity matrix and A and B and I are square matrioss of the same order. Note that only square matrices may have inverses. Before we consider the division or inversion of matrices wa have to introduce the term determinant, Definitions The number associated with a square matrix is known as the determinant of the matrix.

19-1 3

Property 2 If 2 rows or 2 columns are interchanged, the sign of the determinant is changed. Example:

The first and second rows are interchanged.

Property 3 If the corresponding element rows or two columns are identical, then the value of the determinant is zero. Example:

-

The first and 3rd columna ara identical.

Proverty 4 If the corresponding elements of two) rowe or two columnis bear the same ratio, then the value of the determinant is zero. Examples

(a+b+c) |1. 10

ba. 0

0:0 0-0

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