1969-03-27 — Page 22

阿二第張六第日十初月二年西己 WAH KIU YAT PO

報日僑筆

四期星26日七廿月三年九六九一髫公年八十五州民警育教僕费

文中學會考試題預習

現代數學科

MODERN MATHEMATICS (21).

Solutions of the exercises assigned last week

(1) From a matrix 3x5 such that i 1.

A+(B+C7

definition

But each

of addition of matrices

01J

associative property of addition

of reals

Therefore, (A+B)+C = A+(B+C).,

(4) Identity element property

auch analogous to the operations of

Since the operations of matrices are so

Solution:

numbers in algebra, it is suggestive for us to define a matrix such that it

Let

11 12 13 14 15

21 222 223 24 25

be the matrix

*35

813=

Therefore the matrixies

8 9

8 9 10 11

(2) From a matrix of order 2x4 such that.

Solution:

Let the matrix be

811 812 843 816

421 822

According to the condition-

the matrix is,

lve the equation:

Solution:

"herefore, But, given

If A 18

find:1

Solution:

Matrices

Algebra: of matrices

etc.

which is A itself

We shall have a general study of matrix algebra just as we study the algebra of numbers. The matrix algebra is a logical deductive system built upon some basic definitions of equal matrices,

6. addition.

subtraction and multiplication of matrices.

of the sum of two or more matrices, the product of matrices and difference of matrices. Using these definitions we can establish some laws, obeyed by the matrices under the operations of addition and multiplication.

In what follows we shall give the dese

Addition of matrices and difference matrices.

Definition If A and B are two matrices of the same order, then the sum of A and B is the matrix, which is obtained by adding, together the corresponding elements of A and B. denoted by A+B Similarly, the difference between two matrices A and B of the same order is defined to be the matrix which is obtained by subtracting each element of A the corresponding element of B denoted by A-B.

Examples

(a) 2 3 1

_~°~ (-3 3 } ) - ( -3 · - 3 ) - ( 2 = 19)

6. Properties of addition of matrices

(1) Closure property

Obviously the sum of two natrices is again a matrix, which is of the same order of the original ones.

(2) Commutative property.

Let A (aq), ·B= (bij m

mxn

then A+B=B+A. Proof:

A+B = 1 ay

and B+A

41mxn

man

definition of addition of matrices

But each a* 43

the commutative property of addition

of reals.

Therefore, ABB

(3) ASSociative property:

Let A = (aij)uxn • 8 =

Then (A+B+C = A+{B+C). Proof

min

C= (cijmxn

corresponds to the number zero in algebra. Definition: A matrix with all elements to be zeros is called a zero matrix, denoted

If A and O are of the same order, ther

A + 0 = 0 + A = A

Proof:

Each element in the resulting matrix is in the form + 0, which is

aij.

Similarly

Therefore

0

0+ A A

(5) Inverse element property

If A and B are of the same order and each pair of the corresponding elements in the two matrices is a pair of numbers that are additive inverses of each other.

then

SBO and B is called the additive

-inverse of A or A is the additive inverse

of B. Obviously it is always possible. "for each matrix to have an: additive

inverse. This 18 what we call the inverse element property.

Remarka

(1) Since addition of matrices is closed

and associative, there is an identity element 0, and for each matrix there is an additive inverse: we conclude that the set of all conformable

matrices is a group under addition. When two matrices are of the same order they are called conformable for addition

2). Notice that it is the inverse element property of addition of matrices that makes subtraction of matrides possible.

9. Worked examples

(1) Compute

Solution:

2

12-6

4-8

(2) Compute

(36) •

The two matrices are not conformable for addition, for their orders are different, on order being 2x2 and the other 3x3.

(3) Find values x,y a and b that satisfy the

matrix relationship.

20+4

Solution

G

27-8) 4x+6

3b

Since the two matrices":

s are equal, the corresponding elements must be equal.

26

(4) Compute

What is

Solution:

What is

(5) Wha

It ia

10 -1

917

923

10. Exercise for this week:

(1) Give

-3

2

Ba

6 87

Compute the following:

to the element:

'65#2%3A6%3#$$$#546545#LAKI 348#$%

橋

一九六九市中京神学会考試題殖語

英文科

(二十一)

·王淑方

LESSON TWENTY-ONE

-3-69.

SECTION FOUR

TRANSLATION (continued)

EXERCISE 31

Translate the following passage into Chinese:

A small crowd had gathered round the entrance to the park. Chan Sing crossed the road to see what was happening. He found that the centre of attraction was an old man with a performing monkey. The monkey's tricks, he soon discovered, were in no way remarkable so, after throwing a few coins in the direy hat which the man had placed on the pavement, Chan Sing began to】 move off, along with other members of the crowd.

cry.

At this point the man suddenly let out a loud Everyone turned to see what had happened. The man was bending over his monkey, which now lay quite still on the pavement. He picked up the apparently lifeless body

and, holding it close to him, began to weep. young man stepped forward from the crowd and, taking some money from his pocket, dropped it into the hat. Chan Sing and several other people did likewise, unțil the coins in the hat were covered with dollar notes. -Meanwhile, the man continued to hold the dead nopkey if his arms and seemed to take no notice of what was going on about him

EXERCISE 321

Translate the following passage into Cria

party began shortly after nine. Mr. Tang, who lived in the flat below, sighed to himself as he heard the first signs: the steady tramp of

the stairs;

the sound of excited voices as the red, one

another; and the noise of the gramophone, which was tu turned full on. Luckily Mr. Tang had brought some work home from the office, with which he occupied himself for a couple of hours, thus managing to ignore with some. success the party which was going on over his heard. But by elever o'clock he felt tired and was ready to go to bed, though from his experience of previous parties he knew that it was useless trying to get to sleep. He undressed and lay for a while on the bed, trying" co read, but the noise from the room directly above his head did not allow him to concentrate on what he was reading. He found himself reading the same page over. and over again. He then switched off the light and buried his head in the pillow, in a desperate effort to go to sleep. But even so he could not shut out the noise. Finally, after what seemed hours, he switched on the light and looked at his watch it was just, after). midnight.

EXERCISE 33

Translate the following passages into English:

ANSWERS TO PREVIOUS EXERCISES

EXERCISE 28

(1)

(b)

vicious

inflicted

forbears

(d)tines

preserves

ornately essential genteel

(i) prevalent

· (2) The ancient Egyptians used forks for agricultural

purposes.

(3) The three kinds of forks used in the ancient days

are pitchforks, hayforks and digging forks.

སྤྱི་༡ཏུ་སྤྱི。

The word 'ugly' is used to describe the wounds in the sense of serious wound with an unpleasant "look.

(5) The word 'record' means the written account of

matters

(6) The idea of eating with a fork cane into Europe

from the East some two centuries before the rst | record of any table fork in England.

(7) The metal forks in Henry VIII's days were employed

only to pick up Indian preserves called 'ginger' (8) The custom of eating meat with a fork was wide-

spread in Italy by the fifteenth century because the Italians were averse to having their food touched with fingers (which were not clean).

EXERCISE 29

+02x2

make sense

(A - C).

(2) Does the expression

(2 : ;).

Compute

(

) f 2

(4) If A and B are conformable for addition

prove

B

(A+B)

(A+B)+C × (B1]