11.1matrices 11.2determinants 11.3inverses of square matrices chapter summary case study matrices...

11.1Matrices

11.2 Determinants

11.3 Inverses of Square Matrices

Chapter Summary

Case Study

Matrices and Determinants11

P. 2

Team Xproduce 500 pieces of product A, 200 pieces of product B and 350 pieces of product C

Team Yproduce 200 pieces of product A, 400 pieces of product B and 450 pieces of product C

Case StudyCase Study

Contents of product A: 1.5 kg of copper, 0.2 kg of steel product B: 0.6 kg of copper, 1.4 kg of steel product C: 0.8 kg of copper, 1 kg of steel

How to organize and calculate the total amount of copper and steel needed by each team?

Great! Please calculate the total amount of materials needed by each team.

We received an order to produce three kinds of products. Teams X and Y will work together to finish this job.

(1) Amount of copper needed by Team X ?(2) Amount of steel needed by Team X ?(3) Amount of copper needed by Team Y ?(4) Amount of steel needed by Team Y ?

That’s tedious!

P. 3

OrganizationWe can arrange the data in tabular form:

Calculation(1) Amount of copper needed by Team X

Product A Product B Product C

Team X 500 200 350

Team Y 200 400 450

Copper (in kg) Steel (in kg)

Product A 1.5 0.2

Product B 0.6 1.4

Product C 0.8 1

Copper Steel

Team X 1150 kg

Team Y

(500 1.5 200 0.6 350 0.8) kg 1150 kg

(2) Amount of steel needed by Team X ?

(3) Amount of copper needed by Team Y ?

(4) Amount of steel needed by Team Y ?

730 kg

900 kg 1050 kg

1st row

1st row

1st column

1st column

2nd column

2nd column

Case StudyCase Study

P. 4

A rectangular array of numbers arranged in m rows and n columns is called a m n matrix.

11.1 Matrices11.1 Matrices

An m n matrix is represented in the form

mnmmm

n

n

n

aaaa

aaaa

aaaa

aaaa

321

3333231

2232221

1131211

mnmmm

n

n

n

aaaa

aaaa

aaaa

aaaa

321

3333231

2232221

1131211

or

A matrix with m rows and n columns is said to be a matrix of order m n.

The number aij in the ith row and the jth column of a matrix is called an

element or entry.

For example, in the 2 3 matrix , a12 4 and a23

7.

710

541

3rd column

2nd row

mth row

nth column

A. A. IntroductionIntroduction

P. 5

For an a m n matrix,if m 1, it has only 1 row and is called a row matrix;if n 1, it has only 1 column and is called a column matrix.

We should specify the row number first, then the column number.

( 5 4 3 ) is a row matrix of order 1 3.

is a column matrix of order 3 1.

3

4

5

Two matrices are said to be equal if they satisfy the following definition:

Equality of MatricesTwo matrices A (aij)m n and B (bij)m n are equal if

and only if they have the same order and their corresponding elements are equal, i.e.,

aij bij for all i = 1, 2, 3, ... , m and j = 1, 2, 3, ... , n.

11.1 Matrices11.1 MatricesA. A. IntroductionIntroduction

P. 6

Example 11.1T

Solution:

If , find the values of w, x, y and z.

zy

xw

8

0

1086

420

From the definition,w 2, x 4, y 6 and z 10.

11.1 Matrices11.1 MatricesA. A. IntroductionIntroduction

P. 7

Zero MatrixA zero matrix, or a null matrix, is a matrix that all its elements are zero.

For example, is a 2 3 matrix.

000

000

Square MatrixA square matrix is a matrix with the same numbers of rows and columns.

For example, is a square matrix of order 2.

43

21

Notes:

The order of a square matrix is denoted by its number of rows n.

11.1 Matrices11.1 MatricesB. Special Types of MatricesB. Special Types of Matrices

P. 8

For example, is the identity matrix of order 3.

100

010

001

Identity MatrixAn identity matrix of order n, which is denoted by I, is an

n n square matrix with .

ji

jiaij for 0

for 1

An identity matrix is also calleda unit matrix.

11.1 Matrices11.1 MatricesB. Special Types of MatricesB. Special Types of Matrices

P. 9

Some rules on the operations of matrices:

Addition of MatricesSuppose A (aij)m n and B (bij)m n

are two matrices of

order m n. Then the sum of A and B is also an m n matrix C (cij)m n with

cij aij bij, for all i 1, 2, 3, ... , m and j 1, 2, 3, ... , n.

For example, if and , then

40

31A

31

12B

. 71

23

3410

)1(321

BA

Note that the addition of matrices is defined only when the two matrices are of the same order.

11.1 Matrices11.1 MatricesC. Operations of MatricesC. Operations of Matrices

P. 10

Negative of MatricesLet A (aij)m n be an m n matrix. The negative of A,

denoted by A, is the matrix whose elements are the negative of the corresponding elements of A, i.e.,

A (aij)m n, for all i 1, 2, 3, ... , m and j 1, 2, 3, ... ,

n.

For example, if , then .

43

21A

43

21

4)3(

)2(1A

Subtraction of MatricesSuppose A (aij)m n and B (bij)m n

are two matrices of

order m n. The difference of A and B is defined asA B A (B).


P. 11

Example 11.2T

Solution:

Suppose and . Find the matrix Z such that

Y Z X.

31

21X

10

24Y

∵ Y Z X∴ Z X Y

10

24

31

21

10

24

31

21

21

45When summing up matrices, we sum up each pair of the corresponding elements independently.


P. 12

Properties of Matrix AdditionLet A (aij)m n, B (bij)m n and C (cij)m n

be m n

matrices and 0 be the m n zero matrix. Then we have:(a) A B B A (Commutative Law)(b) (A B) C A (B C) (Associative Law)(c) A 0 0 A A(d) A (A) (A) A 0

Proofs of (a) and (b): By the definition of addition of matrices,

A B (aij)m n (bij)m n

(aij bij)m n

(bij aij)m n

(bij)m n (aij)m n

B A

(A B) C (aij bij)m n (cij)m n

[(aij bij) cij]m n

[aij (bij cij)]m n

(aij)m n (bij cij)m n

A (B C)


P. 13

Scalar Multiplication of MatricesThe scalar multiplication of an m n matrix A (aij)m n

and a real number k, which is denoted by kA, is an m n matrix whose elements are the corresponding elements of A multiplied by k, i.e.,

kA (kaij)m n, for all i 1, 2, 3, ... , m and j 1, 2, 3, ... ,

n. For example, .

86

42

43

212

Properties of Scalar MultiplicationLet A and B be two m n matrices and h, k be two real numbers. We have(a) k(A B) kA kB; (Distributive Law)(b) (h k)A hA kA;(c) hkA h(kA) k(hA).


P. 14

Example 11.3T

Solution:

Suppose and . Evaluate 2X 3Y and 4Y 2X.

12

44

23

X

45

24

12

Y

1215

612

36

24

88

46

1411

24

10

45

24

12

3

12

44

23

22X 3Y

12

44

23

2

45

24

12

44Y 2X

24

88

46

1620

816

48

1424

1624

814


P. 15

Multiplication of MatricesLet A (aij)m n be an m n matrix and B (bij)n p be an

n p matrix. The product AB is an m p matrix C (cij)m

p where

cij ai1b1j ai2b2j ... ainbnj ,

for all i 1, 2, 3, …, m and j 1, 2, 3, …, p.

n

kkjikba

1

To understand the process of the multiplication of matrices, students may also refer to the Case Study at the beginning of this chapter.

Notes:

When calculating the product AB, the matrix A should be placed on the left while B is placed on the right.

Multiplication of matrices is non-commutative, i.e., for two matrices A and B, AB BA in general.


P. 16

Suppose and .

024

311A

12

20

41

B

12

20

41

024

311AB

∴ AB is a 2 2 matrix.

)1(02)2()4(4)2(00)2(14

)1(32)1()4(1)2(30)1(11

Also consider the product BA.∵ B is a 3 2 matrix and A

is a 2 3 matrix.∴ BA is a 3 3 matrix.∴ AB BA

∵ A is a 2 3 matrix and B is a 3 2 matrix.

124

17


P. 17

Example 11.4T

Solution:

For each of the following pairs of matrices X and Y, find XY and YX.

(a) , (b) ,

412

314X

52

01

25

Y

313

124X

154

434

224

Y

(a) XY

52

01

25

412

314

540)1(22)2(4)1()1(52

530)1(24)2(3)1()1(54

243

2315


P. 18

Example 11.4T

Solution:


(a) , (b) ,

412

314X

52

01

25

Y

313

124X

154

434

224

Y

(a)

4532)1(5)1(225424031)1(0)1(12041

4235)1(2)1(52245

1432314

23724

412

314

52

01

25

YX


P. 19

Example 11.4T

Solution:


(a) , (b) ,

412

314X

52

01

25

Y

313

124X

154

434

224

Y

(b) XY

154

434

224

313

124

)1(3)4)(1()2(3)5(3)3)(1()2(3)4(3)4)(1()4(3)1(1)4)(2()2(4)5(1)3)(2()2(4)4(1)4)(2()4(4

YX is undefined.

136417920

The number of columns of Y is not equal to the number of rows of X.


P. 20

Even though A 0 and B 0, we still have AB 0:

Suppose , and .

00

01A

10

00B

01

00C

∴ AB 0 does not imply A 0 or B 0.

00

00

10000000

10010001

10

00

00

01AB

∴ AB AC does not imply A 0 or B C 0.

Consider AB AC

The following shows AC 0:

00

00

00001000

00011001

01

00

00

01AC

∵ A 0 and B C.

AB AC 0A(B C) 0


P. 21

Example 11.5T

Solution:

Let . Find a non-zero square matrix B of order 2 such that

(a) AB 0, (b) BA 0.

10

00A

Let , where a, b, c and d are some constants.

dc

baB

(a)

dc

baAB

10

00

dc

00

∴ c d 0

∵ AB 0

00

baB∴

(b)

10

00

dc

baBA

d

b

0

0

∴ b d 0

∵ BA 0

0

0

c

aB∴


P. 22

Properties of Matrix MultiplicationLet h and k be real numbers and A, B and C be matrices such that the following matrix products are defined. We have:(a) (AB)C A(BC); (Associative Law)(b) (i) A(B + C) AB + AC;

(ii) (A + B)C AC + BC;(c) k(AB) (kA)B A(kB);(d) (hA)(kB) (hk)AB;(e) A0 0A 0, where A is a square matrix and 0 is

a zero square matrix; (f) AI IA A, where A is a square matrix and I is

an identity matrix.

(Distributive Law)

Remarks:

The proofs are left for students.


P. 23

Power of Square MatricesFor any square matrix A and any positive integer n, we have

. ... times

n

n AAAAA

For square matrices A and B of same order:

1. (A B)2

(A B)(A B) AA AB BA BB A2 AB BA B2

2. (A B)(A B) AA AB BA BB A2 AB BA B2

In general, (A B)2 A2 2AB B2 and (A B)(A B) A2 B2.

The expressions cannot be reduced to the form we learnt in junior form unless AB BA.


P. 24

Example 11.6T

Solution:

Let .

310

432

031

X

(a) X

2

310432031

310432031

)3(3)4(1)0(0)1(3)3(1)3(0)0(3)2(1)1(0

)3(4)4(3)0(2)1(4)3(3)3(2)0(4)2(3)1(2

)3(0)4)(3()0(1)1(0)3)(3()3(1)0(0)2)(3()1(1

X 2 XX.

X 2 is also a 3 3 matrix.

(a) Find the matrix X

2.(b) Hence, find the matrix 3X

2 2X 4I, where I is the 3 3 identity matrix.

136224741265


P. 25

Example 11.6T

Solution:

Let .

310

432

031

X

(b) 3X

2 2X 4I

100010001

4310432031

2136224741265

3

(a) Find the matrix X

2.(b) Hence, find the matrix 3X

2 2X 4I, where I is the 3 3 identity matrix.

2916664118361217

400040004

620864062

39186722112361815


P. 26

Example 11.7T

Solution:

If , show, by mathematical induction, that

for all positive integers n.

b

aX

0

0

n

nn

b

aX

0

0

For n 1, obviously L.H.S. R.H.S.∴ The proposition is true for n 1.

k

kk

b

aX

0

0Assume the proposition is true for some positive

integers k, that is, .

When n k 1, L.H.S. X k

1

b

a

b

ak

k

0

0

0

0

1

1

0

0k

k

b

a

R.H.S. ∴ The proposition is true for n k 1.

The following shows an outline of solution only. Students should show your workings clearly.

When n k 1, show that

R.H.S. .

1

1

0

0k

k

b

a


P. 27

Transpose of MatrixLet A (aij)m n be an m n matrix. The transpose of

matrix of A, denoted by At or AT, is an n m matrixAt (cij)n m such that

cij aji for all i 1, 2, … n and j 1, 2, …, m.

The transpose of a matrix A is obtained by interchanging the rows and the columns in A, for examples:

642

531 ,

65

43

21tMM

8

9 , )89( tXX


P. 28

Properties of TransposesLet A and B be two m n matrices, we have(a) (At)t A;(b) (A B)t At Bt;(c) (kA)t kAt, where k is any constant.Let A be an m n matrix and B be an n p matrix, we have(d) (AB)t BtAt.

Remarks:

The proofs are left for students.


P. 29

Example 11.8T

Given that and . If (At)2 pAt qI 0, find the

values of p and q.

65

23A

10

01I

Solution:

62

53tA

4618

4519

62

53

62

53)( 2tA

∵ ( At )2 pAt qI 0

00

00

10

01

62

53

4618

4519qp∴

00

00

646218

545319

qpp

pqp

By comparing the corresponding elements of the matrices on both sides, we have

p 9 and q 8 .


P. 30

For an n n square matrix A ,

nnnn

n

n

aaa

aaaaaa

21

22221

11211

nnnn

n

n

aaa

aaaaaa

21

22221

11211

denote the determinant of A by .

Similar to matrices, only determinants of at most order 3 will de discussed.

Determinant of Order 2

For a 2 2 square matrix A , the value of its

determinant, which is denoted by | A| or det A, is defined by

a11a22 a12a21.

a11a22 a12a21 is called the expansion of the determinant.2221

1211

aaaa

2221

1211

aaaa

11.2 Determinants11.2 DeterminantsA. A. IntroductionIntroduction

P. 31

Example 11.9T

If 5, find the value of x.1

25

xx

xx

Solution:

5)2()1)(5(

51

25

xxxx

xx

xx

5256 22 xxxx

0

04

x

x

A. A. IntroductionIntroduction

11.2 Determinants11.2 Determinants

P. 32

To memorize the expansion of the determinant:

333231

232221

131211

aaaaaaaaa

3231

2221

1211

aaaaaa

This rule is only applicable for determinants of order 3.

Notes:

This rule is called the rule of Sarrus.

Determinant of Order 3

For a 3 3 square matrix A , the value of

its determinant is defined by

a11a22a33 a12a23a31 a13a21a32

a13a22a31 a11a23a32 a12a21a33.333231

232221

131211

aaaaaaaaa

333231

232221

131211

aaaaaaaaa


P. 33

Example 11.10T

Solution:

Evaluate the following determinants.

(a) (b)

104

125

312

01

10

10

c

b

a

(a) (2)(2)(1) 1(1)(4) 3(5)(0) 3(2)(4) (2)(1)(0) 1(5)(1)104

125

312

21

(b) a(1)(0) 0(b)(1) 1(0)(c) 1(1)(1) a(b)(c) 0(0)(0)01

10

10

c

b

a

(1 abc)


P. 34

Example 11.11T

Solution:

Let a, b, c, d and e be five distinct numbers. If , prove that

c(ae bd) a e b d.

0

1

11

1

ed

c

ba

0))(1(1)1)(1()1)(1())(1(1

0

1

11

1

eabcdbdace

ed

c

ba

dbeabdaec

dbeabcdace

)(


P. 35

The following shows some of the properties of determinants, which are true for determinants of any order.

For any square matrix A, the determinant of A is equal to that of the transpose of A, i.e., .tAA

If any two rows (or columns) of a matrix are interchanged, the determinant changes sign but its absolute value remains unchanged.

e.g., ; . 333

111

222

333

222

111

cbacbacba

cbacbacba

333

222

111

333

222

111

cabcabcab

cbacbacba

These properties can be verified by expanding of the determinants.

Remarks:

B. Properties of DeterminantsB. Properties of Determinants


P. 36

If all the elements in any one row (or column) of a matrix are multiplied by a factor, then the determinant is just the product of the original determinant and the factor.

e.g., for any k.

333

222

111

333

222

111

333

222

111

cbakckbkacba

cbacbacba

kcbkacbkacbka

For example, if , then5ihgfedcba

255555

ihgfedcba

ihgfedcba

(i) , (ii) .5ihgfedcba

ghidefabc



P. 37

The determinant of a matrix is zero if all the elements in a row (or column) are zero,

i.e., . 000

0000

33

22

11

321

321

bababa

cccbbb

When k 0, we have:

When all the elements are also multiplied by k, we have:

If all the elements of an n n square matrix are multiplied by the same factor, then the resulting determinant is the product of the original determinant and the nth power of the factor,

i.e., .

333

222

1113

333

222

111

cbacbacba

kkckbkakckbkakckbka



P. 38

The determinant of a matrix is zero if the elements of a row (or a column) are proportional to those of another row (or another column),

i.e., if . 0

333

222

111

cbacbacba

3

3

2

2

1

1

b

a

b

a

b

a

In particular, we have:

If any two rows (or columns) of a matrix are equal, the determinant is equal to zero,

i.e., .

333

222

111

333

111

111

0caacaacaa

cbacbacba



P. 39

Consider the result of addition of matrices, we have:

If all the elements in any row (or column) of a matrix can be expressed as the sum of two terms, then the determinant can also be expressed as the sum of the two determinants,

i.e., .

33

22

11

333

222

111

333

222

111

cbrcbqcbp

cbacbacba

cbracbqacbpa

When p, q and r are proportional to the elements of the other row, we have:

If all the elements in a row (or column) of a matrix is added or subtracted by multiples of the other row (or column), then the value of the determinant will remain unchanged,

i.e., for any k. 333

222

111

333

222

212121

cbacbacba

cbacba

kcckbbkaa



P. 40

Finally, for the product of two square matrices, we have:

For any n n square matrices A and B, the product of their determinants is equal to the determinant of the matrix AB, i.e.,

.ABBA

Verification:

Let and . Then .

dcba

A

hgfe

B

dhcfdgcebhafbgae

AB

L.H.S.

R.H.S.

hgfe

dcba

BA

))(())(( dgcebhafdhcfbgae

dhcfdgcebhafbgae

AB

bcfgbcehadfgadeh

fgehbcad

))((

bcfgbcehadfgadeh



P. 41

Example 11.12T

Solution:

Without expanding the determinant, show that . 03710578

479

3710578

479

3110518

4197

202518101

7

101518101

)2(7

0

Take out the common factor 7 from C2

R1 R2 R1;R3 R2 R3

In order to show the value of the determinant equals to zero, we need to show that any two rows or columns are the same.



P. 42

Example 11.13T

Solution:

Without expanding the determinant, show that 3 is a factor of

. (Given that the determinant is non-zero.)241120032

C2 C3 C2

241120032

261130032

221110012

3 Take out the common factor 3 from C2

Since all the elements in the determinant are integers, its value in an integer.∴ 3 is a factor of the given determinant.



P. 43

Consider the determinant .

ihg

fed

cba

The expansion of the determinant aei bfg cdh ceg afh bdi a(ei fh) b( fg di) c(dh eg)

Group the a terms, the b terms and the c terms;Arrange in alphabetical order

a(ei fh) b(di fg) c(dh eg)

hg

edc

ig

fdb

ih

fea

ihg

fed

cba

ihg

fed

cba

ihg

fed

cba

The / sign of each term is determined by the position ofa, b, c as shown below:

C. Evaluation of Determinants of Order 3C. Evaluation of Determinants of Order 3


P. 44

The expansion of the determinant aei bfg cdh ceg afh bdi b( fg di) e(ai cg) h(cd af )

Group the b terms, the e terms and the h terms;Arrange in alphabetical order

b(di fg) e(ai cg) h(af cd)

fd

cah

ig

cae

ig

fdb

ihg

fed

cba

ihg

fed

cba

ihg

fed

cba

The / sign of each term is determined by the position ofb, e, h as shown below:

Consider the determinant .

ihg

fed

cba



P. 45

Summarize the results as follows:

The determinant of order 3 can be expanded along any row or column,

i.e.,

or , etc. fd

cah

ig

cae

ig

fdb

hg

edc

ig

fdb

ih

fea

ihg

fed

cba

For each of the element, minor

corresponding determinant obtained cofactor

product of the minor and the sign of the term

Remarks:

In a determinant of order 3:For each of the elements a, c, e, g and i, cofactor minor;For each of the elements b, d, f and h, cofactor (minor).



P. 46

Example 11.14T

Evaluate the determinant by expanding along

(a) the first row, (b) the third column.369

825471

Solution:

(a) Value of the determinant69

254

39

85)7(

36

821

1(6 48) 7(15 72) 4(30 18) 615

25

71)3(

69

718

69

254

(b) Value of the determinant

4(30 18) 8(6 63) 3(2 35) 615



P. 47

Example 11.15T

Solution:

Show that . Hence evaluate . 001

455

120

2

121

455

240

121

455

240

121

455

240

001

455

120

2

R3 R1 R3

121

455

120

2

121

455

240

∴

001

455

120

2

45

1212

6

)58(2

Expand along R3



P. 48

Example 11.16T

Let and . Find the determinant of AB.

987

654

321

A

123

456

789

B

0

217

214

211

987

654

321

A

0

112

412

712

123

456

789

B

Solution:

0

BAAB∴

Students may try to find the matrix AB first, and then the determinant of AB. However, each of the determinants of A and B can be evaluated more easily in this case.



P. 49

Example 11.17T

Factorize .32

32

32

zzzyyyxxx

Solution:

32

32

32

zzzyyyxxx

2

2

2

111

zzyyxx

xyz

22

22

2

001

xzxzxyxy

xxxyz

22

22

xzxzxyxy

xyz

xzxy

xzxyxyz

11

))((

))()((

))()((

xzzyyxxyz

yzxzxyxyz



P. 50

Example 11.18T

Prove that .0

333333

222222

111111

cbbaacbaaccbaccbba

Solution:

0

333333

222222

111111

cbbaacbaaccbaccbba

3333

2222

1111

000

cbbabaacaccb

C1 C2 C3 C1

0

333333

222222

111111

cbbaacbaaccbaccbba

∴



P. 51

Example 11.19T

Solve the equation .01

11

2

2

2

xx

xxxx

Solution:

01

11

2

2

2

xx

xxxx

01

12

2

32

xx

xxxxx

, where x 0.

01

1100

2

2

3

xxxx

x

01)1( 2

23

xxxx

0))(1( 43 xxx

0)1( 23 xx1x

01100010001

When x 0, the determinant

becomes .

R1 R3 R1



P. 52

For a non-zero real number n,

is a multiplicative inverse of n such that n 1, i.e., n1n 1.n

1

n

1

For matrices, matrix division is not defined.

We can try to find a matrix B such that BA AB I.

Inverse of a MatrixIf square matrices A and B of order n satisfy the relationship AB = BA = I, where I is the identity matrix of order n, then the matrix B is called the inverse of A and denoted by A1, i.e.,

AA1 A1A I.

11.3 Inverses of Square Matrices11.3 Inverses of Square MatricesA. A. IntroductionIntroduction

P. 53

For example, consider

3152

A

2253

Band

1001

3152

2253

1001

2253

3152

BAAB and

∴ B is the inverse of A and A is the inverse of B.

In particular, the inverse of an identity matrix is the identity matrix itself.

1001

1001

1001

100010001

100010001

100010001 Note the following relation in

real numbers:

11 1 1


P. 54

Actually, not all square matrices have their corresponding inverses.

Singular and Non-singular MatricesA square matrix A is said to be non-singular or invertible if and only if its inverse exists. Otherwise, it is said to be singular or non-invertible.

If the inverse of a square matrix exists, then we have:

Uniqueness of InverseThe inverse of a non-singular square matrix is unique.

Proof (using contraction): Suppose B and C are two distinct inverse matrices of A,i.e., AB BA I and AC CA I.

Then B BI

BAC

IC

C, which contradicts to B C.


P. 55

Consider the matrix .

dcba

A

Suppose , i.e., .

hgfe

A 1

1001

hgfe

dcba

1001

dhcfdgcebhafbgae

By comparing the corresponding elements of the matrices on both sides, we have:

bcad

ah

bcad

bf

bcad

cg

bcad

de

determinant of A

transpose of cofactors of A

For a 2 2 square matrix ,

the cofactors of a, b, c and d are d, c, b and a respectively.

dcba


P. 56

For example, if , then

2358

A

8352

85321

t

A

Inverse of a 2 2 matrix

Let . If A is non-singular, then the inverse of A

is given by:

acbd

bcad

dcba

AA

t

1

ofcofactor ofcofactor ofcofactor ofcofactor 11

dcba

A

The determinant of A is 1.


P. 57

Inverse of a 3 3 matrix

Let . If A is non-singular, then the inverse

of A is given by:

ihgfedcba

A

t

edba

fdca

fecb

hgba

igca

ihcb

hged

igfd

ihfe

AA

11

cofactor

transpose of cofactors of A


P. 58

In general, the inverse of a non-singular matrix A contains the

factor .A

1

Thus if , the matrix is singular.0A

TheoremA square matrix A is non-singular if an only if . 0A

Proof: ‘if ’: If A is non-singular, then there exists a matrix B such that

AB BA I.

1 IBAABIAB∵ , and

∴ 0A

‘only if ’: If , then we can find the inverse:0A tAA

A ) cof(11

∴ A is non-singular.


P. 59

Example 11.20TFind the inverses of the following matrices.

(a) (b)

3414

C

208644311

D

Solution:

(a) 16

)4(1)3(4

C

t

C

4143

16

11

4

1

16

14

1

16

3


P. 60

Example 11.20TFind the inverses of the following matrices.

(a) (b)

3414

C

208644311

D

Solution:

(b)

12844

112

6431

06431

8

D

16

1

16

1

4

164

3

64

11

16

564

9

64

1

16

1t

D

4411

6431

6431

0811

2831

2031

0844

2864

2064

128

11


P. 61

Example 11.21TLet P be a square matrix such that 2I P P2 0. Prove that P is non-singular and find P1 in terms of P and I.

Solution:

2I P P2 0 P P2 2I

IIPP

)(2

1 P(I P) 2I

∴ P is non-singular and .)(2

11 IPP


P. 62

Properties of InversesLet A and B be two non-singular square matrices of the same order, k be a non-zero real number, and n be a positive integer. Then(a) (A1)1 A;(b) (kA)1 k1A1;(c) (An)1 (A1)n;(d) (At)1 (A1)t;(e) ;(f) (AB)1 B1A1.

11 AA

Proof of (f): ∵ (AB)(B1A1) A(BB1)A1

AIA1

AA1 I and

(B1A1)(AB) B1(A1A)B B1IB B1B I

∴ By definition, (AB)1 B1A1.

11.3 Inverses of Square Matrices11.3 Inverses of Square MatricesB. Properties of InversesB. Properties of Inverses

P. 63

Example 11.22T

Let and .

(a) Find A1 and B1.(b) Hence find (AB2)1 and [(AB)t]1.

5672

A

4312

B

Solution:

(a) 52

)6(7)5(2

A

2675

52

11A

5)3(1)4(2

B

2314

5

11B


P. 64

Example 11.22T

Let and .


5672

A

4312

B

Solution:

(b) (AB2)1 (B2)1A1

(B1)2A1

2675

2314

2314

)52)(5(5

1

2675

718619

1300

1

1404814559

1300

1


P. 65

Example 11.22T

Let and .


5672

A

4312

B

Solution:

[(AB)t]1 [(AB)1]t

(B1A1)t

t

2675

2314

)52(5

1

t

2533014

260

1

2530

314260

1

(b)


P. 66

Example 11.23T

Let .

(a) Find M 2.

(b) Hence find M 1.

3753

M

Solution:

(a)

440044

3753

37532M (b

)IM 442

IMM

44

1

3753

44

144

11 MM


P. 67

Example 11.24T

Let .

105254132

X (a) Find X

1.(b) Hence find Y if YX

.

355033112

Solution:(a)

75432

12412

02513

5

X

215250714135

7

1

t

X

5432

2412

2513

0532

1512

1013

0554

1524

1025

7

11


P. 68

Example 11.24T

(a) Find X

1.(b) Hence find Y if YX

.

355033112

Solution:

Let .

105254132

X

(b) Y (YX)X

1

215250714135

355033112

7

1

1152031227

01421

7

1

)2(3)0(5)1(5)15(3)7(5)3(5)25(3)14(5)5(5)2(0)0(3)1(3)15(0)7(3)3(3)25(0)14(3)5(3)2(1)0(1)1(2)15(1)7(1)3(2)25(1)14(1)5(2

7

1


P. 69

Example 11.25T

Let and .

100211202

X

010121

120Y

Solution:

t

Y

)2(1000110)1(

1

201100

011

010121

120

100211202

201100

0111XYY

(a) Find the matrix Y 1XY.(b) Hence find X 1000.

200010001

The following shows an outline of solution only. Students should show your workings clearly.1

2120

011

10)1(

1212

0

Y(a)


P. 70

Example 11.25T

Let and .

100211202

X

010121

120Y

Solution:

(a) Find the matrix Y 1XY.(b) Hence find X 1000.

(b) Consider (Y 1XY)1000 (Y 1XY)(Y 1XY)(Y 1…) … (… Y)(Y 1XY) Y 1X(I) X(I) … (I) XY Y 1X 1000Y

∴ Y(Y 1XY)1000Y 1 X 1000

∵

1000

10001

200010001

)(

XYY

1

1000

1000

200010001

YYX∴

1000200010001

10022121

220210011000

10011000


P. 71

11.1 Matrices

Chapter Chapter SummarySummary

1. DefinitionAn m n matrix is represented in the form

An m n matrix may also be represented by the symbol (aij)m n or [aij]m n.

mnmmm

n

n

n

aaaa

aaaa

aaaa

aaaa

321

3333231

2232221

1131211

P. 72

11.1 Matrices

2. Operations of MatricesLet A (aij)m n and B (bij)m n be two matrices and k be a real

number.(a) AdditionA B (aij bij)m n, for all i 1, 2, ... , m and j 1, 2, ... ,

n(b) SubtractionA B (aij (1)bij)m n, for all i 1, 2, ... , m and j 1, 2, ... , n

(c) Scalar MultiplicationkA (kaij)m n

(d) TransposeAt

(cij)n m where cij aji, for all i 1, 2, ... , n and j 1, 2, ... ,

m(e) MultiplicationLet A (aij)m n, B (bij)n p and C (cij)m p. If AB C,

then

cij ai1b1j ai2b2j ... ainbnj

n

kkjikba

1


P. 73

1. Determinant of order 2

a11a22 a12a212221

1211

aaaa

2. Determinant of order 3

a11a22a33 a12a23a31 a13a21a32 a13a22a31 a11a23a32 a12a21a33

333231

232221

131211

aaaaaaaaa

3332

232213

3332

232212

3332

232211 aa

aaa

aaaa

aaaaa

a

11.2 Determinants


P. 74

1. DefinitionFor a square matrix A, if there exists a matrix B such that

AB BA I,

then B is called the inverse of A and is denoted by A1.

2. Inverse of a 2 2 matrix:

acbd

bcaddcba 11

3. Inverse of a 3 3 matrix:

t

edba

fdca

fecb

hgba

igca

ihcb

hged

igfd

ihfe

AA

11

11.3 Inverses of Square Matrices


11.1matrices 11.2determinants 11.3inverses of square matrices chapter summary case study matrices...

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