1969-03-20 — Page 14

報日僑華

四期星日十二月三年九六九一座公年八十五國民華中

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其二第張四第 日三初月二年西己L WAH KIU YAT PO

育華僑華

[1 Definition of matricas

上

文中學會考

考試題預習

A rectangular array of real numbers is called a matrix. It is important to notice that a matrix does not have a numerical value, but is only a collection of numbers arranged in a rectangular tabla. Mtrices are denoted by capital letters A, B, C and so on. Some examples of matrices are the following:

0-1

現代數學科

(二十)

李義

MODERN HATHEMATICS (20).

Solutione of the exercises assigned. Last week

(3): Prove that the modular: arithmetic mod 3 undar

addition is isomorphic to the group whose elements are the permutations

1.2

₤1.2.31 HP(312)

71.2

under the operation "followed by n, denoted by lion

Solution:

Addition table of arith- mod 3.

won table ar {e, p, ai y

The one-one correspondence between their

●lements is given as follow.

Order of a mat

A matrix with a row and n columns is said to be a matrix of order a x n. This means that the order

a matrix is given by stating first the number of rowa then the number of columns in the matrix. This the orders of matrices given in section 1 are respectively 2 x 3, 2 x 2, 4 x 1, and 1 × 3.

If a matrix has only one column, it is called ■ } column matrix; and if a matrix has only one row, it is called a row matrix. Thus example (3) is a colum mrix and example (4) is a row matrix. If the number of rows is the same as the number of columns, san the second example above, it is called a square

Given two linear equations in two variables; san form a square matrix with its coefficients. Example: Givan 3x-2y7 and I 33-4, the coefficients of x and y form a squa. 378

Matrix (33)

When we are speaking a square intrix of nex order, we often refer to its order as n rather than; nin. Thus the matrix of the coefficients of the two equations above has the order 2.

Bolution?

32

311 42-3

and

(2)Form a matrix of order 3 x 3 auch that adj = 0.

rifj and aid -1f1-1

Solution : Let the matrix be

1; and all the other entri

Thus the matriz becoMISS

(3) Form a matrix of order 4 x 2 auch that asy

21-4. Solution:

Let the strix-ba

x2

3

Henne their operations are "preserved."

(2) Prove that the set (2,4,6,8} under multiplication

mod 10 is isomorphic to the set {1,3,7,9} under multiplication mod 10.

Solution:

Tables of multiplications

The one-to-one correspondance between the

ia given as below,

3. Notations of matrices

·Capital letters are often used to denote general matrices, The entries in a matrix are of denoted by amall letters with subscripts. Thus, may. represent a matrix h of order 3 x 3 by

11 12

#21 22 23 and a matrix B of order 3x4 by 031 *32 033/

#22023

D32

in general, aij means that it is the entry in the ith row and th colum. Thus #32 is the entry in the third row and second column, and a13 As the entry in the first row and third column. The notation for a matrix may be also abbreviated as follows)

Am (a13), and B- (b1j)

(a1) 3 x 3

Equality of two matrices

Two matrices of the same order are said to be @quel if and only if their corresponding elements are equal. Examples+

Hence

(4) How many entries are there in a 4 x 5 matrix?

A matrix? In auxnatrix? Solution: Inaux5 matrix, there are 5. entries in each row, and there are 4 rows. Therefore there are 4 x 5 or 20 entries in a 4 x 5 matrix, Similarly, there are 3 x 4 or 12 entries in 3 x 4 matrix, and there are m x n entreia în am x`n matrix.

(5) Solve: (§ ~ ~ ) − ( 1 )

Solution::

Since the tuo matricea are equal and they are of the same order 2 x 13 the corresponding entries are equal; hence

2:

(6) If A - B and B-C, what would you say about

8 and C7 Axplain Solution:

x 8

and so on. These show that

and d = 0.

((2) (132)-(334)

means that

the operatio

are #preservad", hence the two

sets are isomorphic under multiplisations

(【3】 Solve: x • 6 (mod 7).

Solution: (3 x 3) x 3 - 2 x 3 • 6; (5-x.

4 x 5 - 63. (6 x 6) 16 - 1 x 6. • Therefore x • 3, œr 5, or 6,

{(4) Barve 1x + 8. 3 imod 1).

Solution: 28

3 +36]

(5) What wḍll be the day of the week 17 days from now,

aastuẳng now is Thursday? Solutions 17 4 4 0 (mod ?】

The day will be Sunday

(6) What month of the year will be 37 months from no

Cassuming now is March?

37 • 3 • 4-(mod 12),

Thelfmonth

will be April

The transpose of a ma trix

Given a matrix A, the transpose of A, denotad

by A is formed by interchanging its rows and columns Examples!

than B

23 ;)

Notice that if A the wx matrix.

Porked examplas

(a)t A

Write the transpose

A. Name the entry.

Name the entry,

Since' A = Band B C then

big and bij ani; Therefore

= cyy (transitive property

Similarly, aqz - ¢12) #21

$22

922.

Therefore A= C (definition of equality of

matrices)

Exercises for the week

(1) Form a matrix of order 3 x 5 such that aij

213.

(2) Form a matrix of order 2 x ↳ such that ai) =)

(3) Solve the equations.

(4)

。。。

Find A15):

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港區考生名列前茅

張建、李錦標本人成卓越

根據做實可署昨

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日推九

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直升機動 臨香港教育問題

·潮州公邀請

可做有公能之招事 超不脚

器,深佈止多,缀校學可學

史者外了研來感

豬/李年制的

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因是難有

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詞

粥鎘

,在學英

絲杈作賽時

下學

理了够的夹入天循著

圣。把方的。合去通

隆問項

及麵

不

不事別再必,墓

注意賽點,及朗新一期藏了(席)