1969-04-24 — Page 24

* = TAT [92 (dz + d22 +α32}]

真四第六第

BADAEĦDREI WAH KNJ YAT

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四期星一日四廿月四年九六九一层公年八十五圈民藝中有教體舉

1969FUNZ

Sal

現代數學科(廿五)

1. Evaluates

MODERN HATHEMATICS (25)

the exercises: 89signed last week:

"21 031)

022 432

423 da

131

33

(20.48,

6.2.

35 21

Solutions

35 21

0 3 9

48

x (2

-121

(2 rows are identical)

+b+c+d a+b+c

i+c a+b+

(a+brord)

LIT

a-b b-o ord}

(arbrord) [b-o ord d-a

lo-d d-s a-bi

ord d-

[(a-b)

(arbrora] [(2-0) f(0-1) {3×3) – (dua)2} - {b=0} {{b-o)(a-b) - (0-8)(d-a)}+ (q-d) {(b-o)(a−a) - (c-d)2]]- ? Complate it yourselves,

Solvet

2+k

Solution:

3.

Prove that

Proofs

(2+k)

(2+k) |

(2+k)(k-1) | |

(2+k)(k-1)(x-1)=0

1,-2.

#31

#21

No13 923 33.

1112131

21

#31 32

11

*12.121313 022 912 + 043 213

13

**12022 + *13 423

82222

21832922

*12032 + 293 033

2333

3122

#32 32

(

(店)

0 1

1. Hence A.A Similarl

833 23

(property 6 and property 7)

(property or a matrix multizlied

by a Boolar)

021 31 d12 022 432

323

33

I is proved.

I can be proved as below.

11 12 13

21 22 23

32 33

- ✩ (§ ~ :) - # ( ¦

Remarks: (1) Although only square matrices have multiplicative inverses, but not all square matrices have multiplicative inverses. If the determinant of a given square matrix A, or ja 1a zero, then the multiplicative inverse of A is not defined. In such case matrix á is called a singular matriz, otherwise A is called a non-singular matrix. Hence only non- singular matrices have inverses. (2) Although multiplication of matrices is not commutative, the multiplication of a square matrix and its inverse is commutative.

5. Application

The inversion of matrices can be employed as means to solve a system of linear equations in two, three, or more variables, The procedure, is illustrated as follows,

Let the system of linear equation. be

2

#132

esented by a matrix

* = [*3 (α 13 **32*33)

4. Worked examples +-

(1) Solve the system

₤3x+4y- 15-J

Solution i

G 1)(3)-(6)

Inverse of (3

(34) 1 (3) 1

(4)

B41

*(3)8-90) + ( -4) (2)

(62) (3) • → (3)-G)

()-G

(2) Solve the systèm

2x +

3

Olutions

()()(22)

+(3-2)(3)() + ( )(2)

~_~) (1)

(금) (주)(3)

(3)-(3).

(3) Solver

2x

Solutions

(1)

0400-0 AGT90100-2 6390.

(-1)

C50-A (0-6)

The above can be

equation as below,

/11 212

21 22 23

31 32 32

00

The inverse of the matrix, which is made up of the coefficients of the equations is

1011 £21 231

(4) Solver

7. Work for this week

(1) Solve:

(4x-38 7 15x+y= 23,

Katrices (continued)

4. Inversion of matricas

The objective of this section is to Learn how find the inverse of a matrix with respect to multiplication, That is, given a aquare natriz we are to obtain the inverse of A, denoted by A such that

-A = 1, where A same order as that of A.

and I are of the

The procedure of finding the multiplicative inverse of a square matrix is illustrated as

follows:

Assume

12131

12 22 32

TA 13 23 33/

Kultiplying both sides of the members in the equation above by the inverse of A, we have

where{A}is the determinant

the coefficients matrix.

(2) Solver

2x 5y

3

13 8.

(3) Salve

+22

2x

2z 1

02?

11 21

12 13

22 32 2333

31 32

£11 £21 31

21 22 23 #32 33

Then there is a matrix galled adjoint of ▲. denoted by adj A, which is defined as below.

Adj ▲ is the matrix which is obtained by repacing each element in the transpose of a given, square matrix A by its fofactor. That 18,

11 *21 #31)

12 22 32

13 23 33/

Then the inverse of

கர்

011 21 31

12 22 32 2.13 23 33/1

(12: 22 32

13 23 33.

Simplifying, we have

chat

(2)

M

21 31

TAT 212 A22032 213 223

R11 21 311 112 222 232 18013: 123 33.

(4) Test whether each or the following satřices

singular or non-singular

(a)

(b)

( (} }) ( ( ) _ " ( 1 )

y+22

3x + y -

23

+22 -4.

The matrix of coefficients is

12

+22

But

-2

Therefore the matrix of the coefficients 18 singular and there is no solution for the system;