Matrix

(element)

(element)

(element)

(element)

(row)

(element)

(row)

12

(element)

(row)

12

(element)

(row)(column)

12

(element)

(row)(column)

12

1

2

3

(element)

(row)(column)

12

1

2

3

1

(element)

(row)(column)

12

1

2

3

1

(element)

1

(element)

1

1

(element)

1

1

(element)

1

1

(element)

1

1

(element)

1

1

(element)

1

1

A ij

23

2323

2323

2323

22

2323

22

2323

22 32

2323

22 32

2323

22 32

13

2323

22 32

1331

2323

22 32

1331

2323

22 32

1331

2

2323

22 32

1331

2

3

2323

22 32

1331

2

3

3

1

1

1

1

1

1

1

O 0 0

1

O 0 0

1

O 0 0

3

(1)

(2)

3

(1)

(2)

3

(1)

(2)

4

(1)

(3)

(2)

(4)

4

(1)

(3)

(2)

(4)

4

(1)

(3)

(2)

(4)

4

(1)

(3)

(2)

(4)

4

(1)

(3)

(2)

(4)

4 3A, -2A

4 3A, -2A

4 3A, -2A

6

(1)

(2)

6

(1)

(2)

6

(1)

(2)

7

(1)

(2)

A =

4 32 1

, B =

3 45 6

, C =

1 23 5

7

(1)

(2)

A =

4 32 1

, B =

3 45 6

, C =

1 23 5

7

(1)

(2)

A =

4 32 1

, B =

3 45 6

, C =

1 23 5

7

(1)

(2)

A =

4 32 1

, B =

3 45 6

, C =

1 23 5

7

(1)

(2)

A =

4 32 1

, B =

3 45 6

, C =

1 23 5

7

(1)

(2)

A =

4 32 1

, B =

3 45 6

, C =

1 23 5

7

(1)

(2)

A =

4 32 1

, B =

3 45 6

, C =

1 23 5

7

(1)

(2)

A =

4 32 1

, B =

3 45 6

, C =

1 23 5

8X

(1)

(2)

A =1 23 4

, B =

4 76 2

8X

(1)

(2)

A =1 23 4

, B =

4 76 2

8X

(1)

(2)

A =1 23 4

, B =

4 76 2

8X

(1)

(2)

A =1 23 4

, B =

4 76 2

8X

(1)

(2)

A =1 23 4

, B =

4 76 2

8X

(1)

(2)

A =1 23 4

, B =

4 76 2

8X

(1)

(2)

A =1 23 4

, B =

4 76 2

2344032460 x y

2344032460 x y

2344032460 x y

2344032460 x y

2344032460 x y

1 32 4

=

21

58

1 32 4

=

21

58

1 32 4

=

21

58

1 32 4

=

21

58

1 32 4

=

21

58

1 32 4

=

21

58

91

2

3

4

2 30 1

12

=42

1 03 1

35=3

4

1 00 1

xy

=xy

1 32 4

=

21

58

91

2

3

4

2 30 1

12

=42

1 03 1

35=3

4

1 00 1

xy

=xy

1 32 4

=

21

58

91

2

3

4

2 30 1

12

=42

1 03 1

35=3

4

1 00 1

xy

=xy

1 32 4

=

21

58

91

2

3

4

2 30 1

12

=42

1 03 1

35=3

4

1 00 1

xy

=xy

1 32 4

=

21

58

1 32 4

=

21

58

1 32 4

1 32 4

=

21

58

1 32 4

12

=

1 32 4

=

21

58

1 32 4

12

56

=

1 32 4

=

21

58

1 32 4

12

56

=

1 32 4

=

21

58

1 32 4

12

56

=

1 32 4

=

21

58

12

56

1 32 4

=

21

58

12

56

10

1

2

3

4

10

1

2

3

4

10

1

2

3

4

10

1

2

3

4

10

1

2

3

4

1 32 4

=

21

58

12

56

1 32 4

=

21

58

12

56

1 32 4

1 32 4

=

21

58

12

56

1 32 4

=

1 32 4

=

21

58

12

56

1 32 4

=

1 32 4

=

21

58

12

56

1 32 4

=

1 32 4

=

21

58

12

56

1 32 4

=

21

58

12

56

1 32 4

=

21

58

12

56

1 32 4

=

21

58

12

56

22

1 32 4

=

21

58

12

56

22

23

1 32 4

=

21

58

12

56

22

23

23

1 32 4

=

21

58

12

56

22

23

23

1 32 4

=

21

58

12

56

22

23

23

1 32 4

=

21

58

12

56

22

23

23

1 32 4

=

21

58

12

56

22

23

23

1 32 4

=

21

58

12

56

22

23

23

1 32 4

=

21

58

12

56

22

23

23

1 32 4

=

21

58

12

56

22

23

23

1 1

1 32 4

=

21

58

12

56

22

23

23

1 1 11

1 32 4

=

21

58

12

56

22

23

23

1 1 11

1 32 4

=

21

58

12

56

22

23

23

1 1 11

1 32 4

=

21

58

12

56

22

23

23

1 32 4

=

21

58

12

56

22

23

23

1 32 4

=

21

58

12

56

22

23

23

23

1 32 4

=

21

58

12

56

22

23

23

232

1 32 4

=

21

58

12

56

22

23

23

232 3

22

23

23

22

23

23

22

23

23

22

23

23

m m

m

m

22

23

23

m m

m

m

22

23

23

m m

m

m

22

23

23

m ml

l

m

m

22

23

23

m ml n

l

m

m

n

22

23

23

m ml n l

l

m

m

n

l

22

23

23

m ml n l n

l

m

m

n

l

n

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c23

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232 3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 +

3

22

23

23

m ml n l n

l

m

m

n

l

n

A B C

c232

c23 = a21b13 + a22b23 + =

mk=1

a2kbk3

3

11

(1)

(2)

3 125=1

3 2 1

214

= 0

11

(1)

(2)

3 125=1

3 2 1

214

= 0

11

(1)

(2)

3 125=1

3 2 1

214

= 0

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

12

1

2

a 0 00 b 00 0 c

p 0 00 q 00 0 r

=ap 0 00 bq 00 0 cr

1 1 10 1 10 0 1

1 2 30 1 20 0 1

=1 3 60 1 30 0 1

Diagonal matrix

Upper triangle matrix

13

1

2

1 11 32 0

4 3 10 2 3

=

4 1 24 9 108 6 2

3 1

2 15 4

=11 1

21

3 4

=6 83 4

3

13

1

2

1 11 32 0

4 3 10 2 3

=

4 1 24 9 108 6 2

3 1

2 15 4

=11 1

21

3 4

=6 83 4

3

13

1

2

1 11 32 0

4 3 10 2 3

=

4 1 24 9 108 6 2

3 1

2 15 4

=11 1

21

3 4

=6 83 4

3

13

1

2

1 11 32 0

4 3 10 2 3

=

4 1 24 9 108 6 2

3 1

2 15 4

=11 1

21

3 4

=6 83 4

3

14A,B,C

B =10

C =

2 1

A =

1 10 2

14A,B,C

B =10

C =

2 1

A =

1 10 2

2 2 2 1 1 2

14A,B,C

B =10

C =

2 1

A =

1 10 2

AB =

1 10 2

10

=10

2 2 2 1 1 2

14A,B,C

B =10

C =

2 1

A =

1 10 2

AB =

1 10 2

10

=10

CA =

2 1

1 10 2

=2 4

2 2 2 1 1 2

14A,B,C

B =10

C =

2 1

A =

1 10 2

AB =

1 10 2

10

=10

CA =

2 1

1 10 2

=2 4

BC =

10

2 1

=2 10 0

2 2 2 1 1 2

14A,B,C

B =10

C =

2 1

A =

1 10 2

AB =

1 10 2

10

=10

CA =

2 1

1 10 2

=2 4

BC =

10

2 1

=2 10 0

CB =

2 1

10

= 2

2 2 2 1 1 2

14A,B,C

B =10

C =

2 1

A =

1 10 2

AB =

1 10 2

10

=10

CA =

2 1

1 10 2

=2 4

BC =

10

2 1

=2 10 0

CB =

2 1

10

= 2

2 2 2 1 1 2

1 (kA)B = A(kB) = k(AB)

(AB)C = A(BC)

(A+B)C = AC +BCA(B + C) = AB +AC

2

3

AB = BA

AB=BA

AB=O

AE=EA=A EIdentityn

A =

1 00 2

, B =

0 11 0

A =

1 00 2

, B =

0 11 0

AB =

1 00 2

0 11 0

A =

1 00 2

, B =

0 11 0

AB =

1 00 2

0 11 0

=0 12 0

A =

1 00 2

, B =

0 11 0

AB =

1 00 2

0 11 0

BA =

0 11 0

1 00 2

=0 12 0

A =

1 00 2

, B =

0 11 0

AB =

1 00 2

0 11 0

BA =

0 11 0

1 00 2

=0 12 0

=0 21 0

A =

1 00 2

, B =

0 11 0

AB =

1 00 2

0 11 0

BA =

0 11 0

1 00 2

=0 12 0

=0 21 0

A =1 00 2

, B =

0 11 0

AB =

1 00 2

0 11 0

BA =

0 11 0

1 00 2

=0 12 0

=0 21 0

A =1 00 2

, B =

0 11 0

AB =

1 00 2

0 11 0

BA =

0 11 0

1 00 2

=0 12 0

=0 21 0

A =1 00 a

, B =

0 11 0

16

a

A =1 00 2

, B =

0 11 0

AB =

1 00 2

0 11 0

BA =

0 11 0

1 00 2

=0 12 0

=0 21 0

A =1 00 a

, B =

0 11 0

16

a a = 1

AO = OA = O

AO = OA = O

A = OB = O

or

AO = OA = O

AB = BA = OA = OB = O

or

AO = OA = O

AB = BA = OA = OB = O

or

AO = OA = O

AB = BA = OA = OB = O

or

A =1 23 6

, B =

2 61 3

AO = OA = O

AB = BA = OA = OB = O

or

A =1 23 6

, B =

2 61 3

AB =1 23 6

2 61 3

AO = OA = O

AB = BA = OA = OB = O

or

A =1 23 6

, B =

2 61 3

AB =1 23 6

2 61 3

=0 00 0

AO = OA = O

AB = BA = OA = OB = O

or

A =1 23 6

, B =

2 61 3

AB =1 23 6

2 61 3

=0 00 0

AO = OA = O

AB = BA = OA = OB = O

or

A =1 23 6

, B =

2 61 3

AB =1 23 6

2 61 3

=0 00 0

AE = EA = A

1 00 1

1 0 00 1 00 0 1

1 0

Elementary MatrixIdentity Matrix

AE = EA = A1 00 1

1 0 00 1 00 0 1

1 0

Elementary MatrixIdentity Matrix

AE = EA = A1 00 1

1 0 00 1 00 0 1

1 0

1 23 4

1 00 1

Elementary MatrixIdentity Matrix

AE = EA = A1 00 1

1 0 00 1 00 0 1

1 0

1 23 4

1 00 1

1 0 00 1 00 0 1

1 2 34 5 67 8 9

Elementary MatrixIdentity Matrix

AX = XA = E X

Inverse matrix

determinant

AX = XA = E X A1

Inverse matrix

determinant

AX = XA = E X A1A1 A1

Inverse matrix

determinant

AX = XA = E X A1A1 A1

A =a bc d

A1 =1

ad bc

d bc a

ad bc = 0

A1

Inverse matrix

determinant

AX = XA = E X A1A1 A1

A =a bc d

A1 =1

ad bc

d bc a

ad bc = 0

A1

= ad bc

Inverse matrix

determinant

AX = XA = E X A1A1 A1

A =a bc d

A1 =1

ad bc

d bc a

ad bc = 0

A1

= ad bc

Inverse matrix

determinantdetA

A =a bc d

, X =

p qr s

, E =

1 00 1

AX = E

A =a bc d

, X =

p qr s

, E =

1 00 1

a bc d

p qr s

=1 00 1

AX = E

A =a bc d

, X =

p qr s

, E =

1 00 1

a bc d

p qr s

=1 00 1

ap+ br = 1 1cp+ dr = 0 2

AX = E

A =a bc d

, X =

p qr s

, E =

1 00 1

a bc d

p qr s

=1 00 1

ap+ br = 1 1cp+ dr = 0 2(ad bc)p = d 1 d 2 b 1cp+ dr = 0 2

AX = E

A =a bc d

, X =

p qr s

, E =

1 00 1

a bc d

p qr s

=1 00 1

ap+ br = 1 1cp+ dr = 0 2(ad bc)p = d 1 d 2 b 1cp+ dr = 0 2p = dadbc 1 (ad bc) 1cp+ dr = 0 2

ap+ br = 1 1cp+ dr = 0 2(ad bc)p = d 1 d 2 b 1cp+ dr = 0 2p = dadbc 1 (ad bc) 1cp+ dr = 0 2

ap+ br = 1 1cp+ dr = 0 2(ad bc)p = d 1 d 2 b 1cp+ dr = 0 2p = dadbc 1 (ad bc) 1cp+ dr = 0 2p = dadbc 1r = cadbc ( 2 1 c) d

ap+ br = 1 1cp+ dr = 0 2(ad bc)p = d 1 d 2 b 1cp+ dr = 0 2p = dadbc 1 (ad bc) 1cp+ dr = 0 2p = dadbc 1r = cadbc ( 2 1 c) d

ap+ br = 1 1cp+ dr = 0 2(ad bc)p = d 1 d 2 b 1cp+ dr = 0 2p = dadbc 1 (ad bc) 1cp+ dr = 0 2p = dadbc 1r = cadbc ( 2 1 c) d

ap+ br = 1 1cp+ dr = 0 2(ad bc)p = d 1 d 2 b 1cp+ dr = 0 2p = dadbc 1 (ad bc) 1cp+ dr = 0 2p = dadbc 1r = cadbc ( 2 1 c) d

ap+ br = 1 1cp+ dr = 0 2(ad bc)p = d 1 d 2 b 1cp+ dr = 0 2p = dadbc 1 (ad bc) 1cp+ dr = 0 2p = dadbc 1r = cadbc ( 2 1 c) d

1

adbc 00 1

d b0 1

a bc d

ap+ br = 1 1cp+ dr = 0 2(ad bc)p = d 1 d 2 b 1cp+ dr = 0 2p = dadbc 1 (ad bc) 1cp+ dr = 0 2p = dadbc 1r = cadbc ( 2 1 c) d

1

adbc 00 1

d b0 1

a bc d

1 0 cd 1d

1

adbc 00 1

d b0 1

a bc d

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

13

1 42 5

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

13

1 42 5

24

1

2

3

4

2 23 4

2 81 4

5 42 1

0 11 0

12

4 23 2

13

1 42 5

0 11 0

25a

A =

a a+ 3a+ 4 a+ 1

25a

A =

a a+ 3a+ 4 a+ 1

25a

A =

a a+ 3a+ 4 a+ 1

a(a+ 1) (a+ 3)(a+ 4) = 0

25a

A =

a a+ 3a+ 4 a+ 1

a(a+ 1) (a+ 3)(a+ 4) = 0

25a

A =

a a+ 3a+ 4 a+ 1

a(a+ 1) (a+ 3)(a+ 4) = 0

25a

A =

a a+ 3a+ 4 a+ 1

a(a+ 1) (a+ 3)(a+ 4) = 0 2(a2 + a 6) = 0

25a

A =

a a+ 3a+ 4 a+ 1

a(a+ 1) (a+ 3)(a+ 4) = 0 2(a2 + a 6) = 0 2(a+ 3)(a 2) = 0

25a

A =

a a+ 3a+ 4 a+ 1

a(a+ 1) (a+ 3)(a+ 4) = 0 2(a2 + a 6) = 0 2(a+ 3)(a 2) = 0

a = 3, 2

A =2 51 3

, B =

6 12 3

X,Y

AX = B

A =2 51 3

, B =

6 12 3

X,Y

AX = B

AX = B A1

A =2 51 3

, B =

6 12 3

X,Y

AX = B

AX = B A1

A1AX = A1B

A =2 51 3

, B =

6 12 3

X,Y

AX = B

AX = B A1

A1AX = A1B EX = A1B

A =2 51 3

, B =

6 12 3

X,Y

AX = B

AX = B A1

A1AX = A1B EX = A1B X = A1B

A =2 51 3

, B =

6 12 3

X,Y

AX = B

AX = B A1

A1AX = A1B EX = A1B X = A1B

=

3 51 2

6 12 3

=

8 122 5

A =2 51 3

, B =

6 12 3

X,YY A = B

A =2 51 3

, B =

6 12 3

X,Y

Y A = B A1

Y A = B

A =2 51 3

, B =

6 12 3

X,Y

Y A = B A1

Y A = B

Y AA1 = BA1

A =2 51 3

, B =

6 12 3

X,Y

Y A = B A1

Y A = B

Y AA1 = BA1 Y E = BA1

A =2 51 3

, B =

6 12 3

X,Y

Y A = B A1

Y A = B

Y AA1 = BA1 Y E = BA1

Y = BA1

A =2 51 3

, B =

6 12 3

X,Y

Y A = B A1

Y A = B

=6 12 3

3 51 2

=17 283 4

Y AA1 = BA1 Y E = BA1

Y = BA1

26

X,Y

AX = B

A =4 25 3

, B =

2 41 1

(1) (2) Y A = B

26

X,Y

AX = B

A =4 25 3

, B =

2 41 1

(1) (2) Y A = B

(1) X = A1B

26

X,Y

AX = B

A =4 25 3

, B =

2 41 1

(1) (2) Y A = B

(1) X = A1B

= 12

3 25 4

2 41 1

26

X,Y

AX = B

A =4 25 3

, B =

2 41 1

(1) (2) Y A = B

(1) X = A1B

= 12

3 25 4

2 41 1

=2 53 8

26

X,Y

AX = B

A =4 25 3

, B =

2 41 1

(1) (2) Y A = B

(1) X = A1B (2) Y = BA1

= 12

3 25 4

2 41 1

=2 53 8

26

X,Y

AX = B

A =4 25 3

, B =

2 41 1

(1) (2) Y A = B

(1) X = A1B (2) Y = BA1

= 12

3 25 4

2 41 1

= 1

2

2 41 1

3 25 4

=2 53 8

26

X,Y

AX = B

A =4 25 3

, B =

2 41 1

(1) (2) Y A = B

(1) X = A1B (2) Y = BA1

= 12

3 25 4

2 41 1

= 1

2

2 41 1

3 25 4

=2 53 8

=13 64 3

(AB)1 = B1A1

(AB)1 = B1A1

A

(AB)1 = B1A1

A AX = EXA = E

(AB)1 = B1A1

A AX = EXA = E

OK

27

(1) (2)

A =3 51 2

, B =

1 12 1

A1 B1 (3) (AB)1

27

(1) (2)

A =3 51 2

, B =

1 12 1

A1 B1 (3) (AB)1

(1)

27

(1) (2)

A =3 51 2

, B =

1 12 1

A1 B1 (3) (AB)1

(1)

(2)

27

(1) (2)

A =3 51 2

, B =

1 12 1

A1 B1 (3) (AB)1

(1)

(2)

(3)

27

(1) (2)

A =3 51 2

, B =

1 12 1

A1 B1 (3) (AB)1

(1)

(2)

(3)

27

(1) (2)

A =3 51 2

, B =

1 12 1

A1 B1 (3) (AB)1

(1)

(2)

(3)

28 A = O A2 = O A

28 A = O A2 = O A

A

28 A = O A2 = O A

A

A2 = O A1

28 A = O A2 = O A

A

A2 = O A1

28 A = O A2 = O A

A

A2 = O A1

28 A = O A2 = O A

A

A2 = O A1

28 A = O A2 = O A

A

A2 = O A1

A = O

28 A = O A2 = O A

A

A2 = O A1

A = O A

1

1

1

A

1

A

1

A

1

A

1

A

1

A

1

A

1

A

1

A

AX = B

AX = B A

AX = B A

AX = B A

AX = B A

AX = B A

30

1

2

3

4

30

1

2

3

4

30

1

2

3

4

30

1

2

3

4

30

1

2

3

4

30

1

2

3

4

3x+ y = 16x+ 2y = 2

3x+ y = 16x+ 2y = 3

3x+ y = 16x+ 2y = 2

3x+ y = 16x+ 2y = 3

3 16 2

3x+ y = 16x+ 2y = 2

3x+ y = 16x+ 2y = 3

3 16 2

= 3 2 1 6 = 0

3x+ y = 16x+ 2y = 2

3x+ y = 16x+ 2y = 3

3 16 2

= 3 2 1 6 = 0

3x+ y = 16x+ 2y = 2

3x+ y = 16x+ 2y = 3

3 16 2

= 3 2 1 6 = 0

2

3x+ y = 16x+ 2y = 2

3x+ y = 16x+ 2y = 3

3 16 2

= 3 2 1 6 = 0

2

3x+ y = 16x+ 2y = 2

3x+ y = 16x+ 2y = 3

3 16 2

= 3 2 1 6 = 0

2

2

3x+ y = 16x+ 2y = 2

3x+ y = 16x+ 2y = 3

3 16 2

= 3 2 1 6 = 0

2

2

3x+ y = 16x+ 2y = 2

3x+ y = 16x+ 2y = 3

3 16 2

= 3 2 1 6 = 0

2

2

2

31

1

2

x 2y = 32x+ 4y = 6

x 2y = 32x+ 4y = 1

31

1

2

x 2y = 32x+ 4y = 6

x 2y = 32x+ 4y = 1

x 2y = 3

31

1

2

x 2y = 32x+ 4y = 6

x 2y = 32x+ 4y = 1

x 2y = 3

ax+ by = 0cx+ dy = 0

(x, y) = (0, 0)

ax+ by = 0cx+ dy = 0

(x, y) = (0, 0)

(x, y) = (0, 0)

ax+ by = 0cx+ dy = 0

(x, y) = (0, 0)

(x, y) = (0, 0)

ax+ by = 0cx+ dy = 0

(x, y) = (0, 0)

(x, y) = (0, 0)

a bc d

ax+ by = 0cx+ dy = 0

(x, y) = (0, 0)

(x, y) = (0, 0)

a bc d

ax+ by = 0cx+ dy = 0

(x, y) = (0, 0)

(x, y) = (0, 0)

a bc d

32ax+ 2(a+ 1)y = 0(a 1)x+ (a+ 1)y = 0

(x, y) = (0, 0) a

32ax+ 2(a+ 1)y = 0(a 1)x+ (a+ 1)y = 0

(x, y) = (0, 0) a

32ax+ 2(a+ 1)y = 0(a 1)x+ (a+ 1)y = 0

(x, y) = (0, 0) a

a(a+ 1) 2(a+ 1)(a 1) = 0

32ax+ 2(a+ 1)y = 0(a 1)x+ (a+ 1)y = 0

(x, y) = (0, 0) a

a(a+ 1) 2(a+ 1)(a 1) = 0 (a+ 1){a 2(a 1)} = 0

32ax+ 2(a+ 1)y = 0(a 1)x+ (a+ 1)y = 0

(x, y) = (0, 0) a

a(a+ 1) 2(a+ 1)(a 1) = 0 (a+ 1){a 2(a 1)} = 0 (a+ 1)(a 2) = 0

32ax+ 2(a+ 1)y = 0(a 1)x+ (a+ 1)y = 0

(x, y) = (0, 0) a

a(a+ 1) 2(a+ 1)(a 1) = 0 (a+ 1){a 2(a 1)} = 0 (a+ 1)(a 2) = 0 a = 1, 2

n1

2=

3

A2 (a+ d)A+ (ad bc)E = O

n

(A+B)2 = A2 + 2AB +B2

n

(A+B)2 = A2 + 2AB +B2

(A+B)2 = (A+B)(A+B)

= A2 +AB +BA+B2

n

(A+B)2 = A2 + 2AB +B2

(A+B)2 = (A+B)(A+B)

= A2 +AB +BA+B2

n

(A+B)2 = A2 + 2AB +B2

(A+B)2 = (A+B)(A+B)

= A2 +AB +BA+B2

n

(A+B)2 = A2 + 2AB +B2

(A+B)2 = (A+B)(A+B)

= A2 +AB +BA+B2

n

(A+B)2 = A2 + 2AB +B2

(A+B)2 = (A+B)(A+B)

= A2 +AB +BA+B2

n

(A+B)2 = A2 + 2AB +B2

(A+B)2 = (A+B)(A+B)

= A2 +AB +BA+B2

OK

n

(A+B)2 = A2 + 2AB +B2

(A+B)2 = (A+B)(A+B)

= A2 +AB +BA+B2

OK

n1

2=

3

A2 (a+ d)A+ (ad bc)E = O

n1

n1

n1

A =1 10 1

n1

A =1 10 1

A2 =1 10 1

1 10 1

n1

A =1 10 1

A2 =1 10 1

1 10 1

=1 20 1

n1

A =1 10 1

A2 =1 10 1

1 10 1

=1 20 1

A3 = A2A =1 20 1

1 10 1

n1

A =1 10 1

A2 =1 10 1

1 10 1

=1 20 1

=1 30 1

A3 = A2A =

1 20 1

1 10 1

A =1 10 1

A2 =1 10 1

1 10 1

=1 20 1

=1 30 1

A3 = A2A =

1 20 1

1 10 1

A =1 10 1

A2 =1 10 1

1 10 1

=1 20 1

=1 30 1

A3 = A2A =

1 20 1

1 10 1

An =1 n0 1

A =1 10 1

A2 =1 10 1

1 10 1

=1 20 1

=1 30 1

A3 = A2A =

1 20 1

1 10 1

An =1 n0 1

A =1 10 1

A2 =1 10 1

1 10 1

=1 20 1

=1 30 1

A3 = A2A =

1 20 1

1 10 1

An =1 n0 1

A =1 10 1

A =1 10 1

=

1 00 1

+0 10 0

A =1 10 1

=

1 00 1

+0 10 0

E

A =1 10 1

=

1 00 1

+0 10 0

E

A =1 10 1

=

1 00 1

+0 10 0

BE

A =1 10 1

=

1 00 1

+0 10 0

BE

An = (E +B)n

A =1 10 1

=

1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

A =1 10 1

=

1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

A =1 10 1

=

1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

A =1 10 1

=1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

A =1 10 1

=1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

B2 =0 10 0

0 10 0

=0 00 0

A =1 10 1

=1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

B2 =0 10 0

0 10 0

=0 00 0

A =1 10 1

=1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

Bn = O (n 2)

B2 =0 10 0

0 10 0

=0 00 0

A =1 10 1

=1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

Bn = O (n 2)

B2 =0 10 0

0 10 0

=0 00 0

A =1 10 1

=1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

Bn = O (n 2)

B2 =0 10 0

0 10 0

=0 00 0

A =1 10 1

=1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

Bn = O (n 2)

B2 =0 10 0

0 10 0

=0 00 0

An = E + nB =1 n0 1

A =1 10 1

=1 00 1

+0 10 0

BE

An = (E +B)n

= En +n C1En1B +n C2En2B2 + +Bn

Bn = O (n 2)

B2 =0 10 0

0 10 0

=0 00 0

An = E + nB =1 n0 1

21 A =1 11 1

21 A =1 11 1

A2 =

1 11 1

1 11 1

21 A =1 11 1

A2 =

1 11 1

1 11 1

=0 22 0

21 A =1 11 1

A2 =

1 11 1

1 11 1

=0 22 0

A4 =

0 22 0

0 22 0

21 A =1 11 1

A2 =

1 11 1

1 11 1

=0 22 0

=4 00 4

A4 =

0 22 0

0 22 0

21 A =1 11 1

A2 =

1 11 1

1 11 1

=0 22 0

=4 00 4

A4 =

0 22 0

0 22 0

= 4E

21 A =1 11 1

A2 =

1 11 1

1 11 1

=0 22 0

=4 00 4

A4 =

0 22 0

0 22 0

A6 = A2A4 = A2(4E) = 4A2 =

0 88 0

= 4E

21 A =1 11 1

A2 =

1 11 1

1 11 1

=0 22 0

=4 00 4

A4 =

0 22 0

0 22 0

A6 = A2A4 = A2(4E) = 4A2 =

0 88 0

= 4E

An ?

A =a 00 b

A =a 00 b

An =an 00 bn

A =a 00 b

An =an 00 bn

A =a 00 b

An =an 00 bn

n1

2=

3

A2 (a+ d)A+ (ad bc)E = O

n1

2=

3

A2 (a+ d)A+ (ad bc)E = O

A2 (a+ d)A+ (ad bc)E = OA =

a bc d

A2 (a+ d)A+ (ad bc)E = OA =

a bc d

A2 =a bc d

a bc d

=a2 + bc ab+ bdac+ cd bc+ d2

A2 (a+ d)A+ (ad bc)E = OA =

a bc d

A2 =a bc d

a bc d

=a2 + bc ab+ bdac+ cd bc+ d2

A2 (a+ d)A+ (ad bc)E = OA =

a bc d

A2 =a bc d

a bc d

=a2 + bc ab+ bdac+ cd bc+ d2

(a+ d)A = (a+ d)

a bc d

=a2 ad ab bdac cd ad d2

A2 (a+ d)A+ (ad bc)E = OA =

a bc d

A2 =a bc d

a bc d

=a2 + bc ab+ bdac+ cd bc+ d2

(a+ d)A = (a+ d)

a bc d

=a2 ad ab bdac cd ad d2

A2 (a+ d)A+ (ad bc)E = OA =

a bc d

A2 =a bc d

a bc d

=a2 + bc ab+ bdac+ cd bc+ d2

(ad bc)E = (ad bc)1 00 1

=ad bc 0

0 ad bc

(a+ d)A = (a+ d)a bc d

=a2 ad ab bdac cd ad d2

A2 (a+ d)A+ (ad bc)E = OA =

a bc d

A2 =a bc d

a bc d

=a2 + bc ab+ bdac+ cd bc+ d2

(ad bc)E = (ad bc)1 00 1

=ad bc 0

0 ad bc

(a+ d)A = (a+ d)a bc d

=a2 ad ab bdac cd ad d2

n

A =1 12 2

A2, A3, A4

n

A =1 12 2

A2, A3, A4

n

A =1 12 2

A2, A3, A4

A2 3A = O

n

A =1 12 2

A2, A3, A4

A2 3A = O A2 = 3A

n

A =1 12 2

A2, A3, A4

A2 3A = O A2 = 3AA3 = A2A = 3A2 = 9A

n

A =1 12 2

A2, A3, A4

A2 3A = O A2 = 3AA3 = A2A = 3A2 = 9A

A4 = A3A = 9A2 = 27A

n

A =1 12 2

A2, A3, A4

A2 3A = O A2 = 3AA3 = A2A = 3A2 = 9A

A4 = A3A = 9A2 = 27A

An = 3n1A

n

A =1 12 2

A2, A3, A4

A2 3A = O A2 = 3AA3 = A2A = 3A2 = 9A

A4 = A3A = 9A2 = 27A

An = 3n1A

n

A =1 12 2

A2, A3, A4

A2 3A = O A2 = 3AA3 = A2A = 3A2 = 9A

A4 = A3A = 9A2 = 27A

An = 3n1A

22

A =4 81 2

A2, A4, A8

22

A =4 81 2

A2, A4, A8

22

A =4 81 2

A2, A4, A8

A2 2A = O

22

A =4 81 2

A2, A4, A8

A2 2A = O A2 = 2A

22

A =4 81 2

A2, A4, A8

A2 2A = O A2 = 2AA4 = (2A)2 = 4A2 = 8A

22

A =4 81 2

A2, A4, A8

A2 2A = O A2 = 2AA4 = (2A)2 = 4A2 = 8A

A8 = (8A)2 = 64A2 = 128A

22

A =4 81 2

A2, A4, A8

A2 2A = O A2 = 2AA4 = (2A)2 = 4A2 = 8A

A8 = (8A)2 = 64A2 = 128A

An = 2n1A

n1

2=

3

A2 (a+ d)A+ (ad bc)E = O

n1

2=

3

A2 (a+ d)A+ (ad bc)E = O

n1

2=

3

A2 (a+ d)A+ (ad bc)E = O

A =2 11 2

, X =

1x

AX = kX(1) k, x

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX (A kE)X = O 1

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

X = O(A kE)X = O 1

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

X = O (A kE)X = O 1

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

X = O (A kE) = 0(A kE)X = O 1

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

X = O (A kE) = 0

(A kE)X = O 1

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(A kE)X = O 1

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(A kE)X = O 1

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(2 k)2 1 = 0

(A kE)X = O 1

A =2 11 2

, X =

1x

AX = kX(1) k, x

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(2 k)2 1 = 0 k 2 = 1 k = 1, 3

(A kE)X = O 1

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(2 k)2 1 = 0 k 2 = 1 k = 1, 3

(A kE)X = O 1

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(2 k)2 1 = 0 k 2 = 1 k = 1, 3

(A kE)X = O 1

(i) k = 1

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(2 k)2 1 = 0 k 2 = 1 k = 1, 3

(A kE)X = O 1

(i) k = 1 1 11 1

1x

= O

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(2 k)2 1 = 0 k 2 = 1 k = 1, 3

(A kE)X = O 1

(i) k = 1 1 11 1

1x

= O x = 1

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(2 k)2 1 = 0 k 2 = 1 k = 1, 3

(A kE)X = O 1

(i) k = 1 1 11 1

1x

= O x = 1

(ii) k = 3

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(2 k)2 1 = 0 k 2 = 1 k = 1, 3

(A kE)X = O 1

(i) k = 1 1 11 1

1x

= O

1 11 1

1x

= O

x = 1

(ii) k = 3

AX = kEX

X = O (A kE) = 0A kE =

2 11 2

k 00 k

=2 k 11 2 k

(2 k)2 1 = 0 k 2 = 1 k = 1, 3

(A kE)X = O 1

(i) k = 1 1 11 1

1x

= O

1 11 1

1x

= O x = 1

x = 1

(ii) k = 3

(i) k = 1 1 11 1

1x

= O

1 11 1

1x

= O x = 1

x = 1

(ii) k = 3

(i) k = 1 1 11 1

1x

= O

1 11 1

1x

= O x = 1

x = 1

(ii) k = 3

P =1 1x1 x2

P1AP

(2) (1) x x1, x2 (x1 < x2)

(i) k = 1 1 11 1

1x

= O

1 11 1

1x

= O x = 1

x = 1

(ii) k = 3

P =1 1x1 x2

P1AP

(2) (1) x x1, x2 (x1 < x2)

P =1 11 1

(i) k = 1 1 11 1

1x

= O

1 11 1

1x

= O x = 1

x = 1

(ii) k = 3

P =1 1x1 x2

P1AP

(2) (1) x x1, x2 (x1 < x2)

P =1 11 1

(i) k = 1 1 11 1

1x

= O

1 11 1

1x

= O x = 1

x = 1

(ii) k = 3

P =1 1x1 x2

P1AP

(2) (1) x x1, x2 (x1 < x2)

P =1 11 1

P1 =

12

1 11 1

(i) k = 1 1 11 1

1x

= O

1 11 1

1x

= O x = 1

x = 1

(ii) k = 3

P =1 1x1 x2

P1AP

(2) (1) x x1, x2 (x1 < x2)

P =1 11 1

P1 =

12

1 11 1

P1AP =

12

1 11 1

2 11 2

1 11 1

(i) k = 1 1 11 1

1x

= O

1 11 1

1x

= O x = 1

x = 1

(ii) k = 3

P =1 1x1 x2

P1AP

(2) (1) x x1, x2 (x1 < x2)

P =1 11 1

P1 =

12

1 11 1

P1AP =

12

1 11 1

2 11 2

1 11 1

=1 00 3

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

n

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

(P1AP )nn

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

(P1AP )n = P1AP P1AP P1APn

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

(P1AP )n = P1AP P1AP P1APEn

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

(P1AP )n = P1AP P1AP P1APE En

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

(P1AP )n = P1AP P1AP P1APE E En

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

(P1AP )n = P1AP P1AP P1AP

n

E E En

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

(P1AP )n = P1AP P1AP P1AP

= P1AnP n

E E En

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

=1 00 3n

(P1AP )n = P1AP P1AP P1AP

= P1AnP n

E E En

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

=1 00 3n

(P1AP )n = P1AP P1AP P1AP

= P1AnP n

E E En

P1AnP =1 00 3n

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

=1 00 3n

(P1AP )n = P1AP P1AP P1AP

= P1AnP n

E E En

P1AnP =1 00 3n

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

=1 00 3n

(P1AP )n = P1AP P1AP P1AP

= P1AnP n

E E En

P1AnP =1 00 3n

An = P

1 00 3n

P1

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

=1 00 3n

(P1AP )n = P1AP P1AP P1AP

= P1AnP n

E E En

P1AnP =1 00 3n

An = P

1 00 3n

P1 =

12

1 11 1

1 00 3n

1 11 1

P1AP =12

1 11 1

2 11 2

1 11 1

=1 00 3

=1 00 3n

(P1AP )n = P1AP P1AP P1AP

= P1AnP n

E E En

P1AnP =1 00 3n

An = P

1 00 3n

P1 =

12

1 11 1

1 00 3n

1 11 1

=12

3n + 1 3n 13n 1 3n + 1

A =2 11 2

, X =

1x

AX = kX(1) k, x

A =2 11 2

, X =

1x

AX = kX(1) k, x

A

11= 1

11

A =2 11 2

, X =

1x

AX = kX(1) k, x

A

11= 1

11

A

11

= 3

11

A =2 11 2

, X =

1x

AX = kX(1) k, x

A

11= 1

11

A

11

= 3

11

A =2 11 2

, X =

1x

AX = kX(1) k, x

A

11= 1

11

A

11

= 3

11

A =2 11 2

, X =

1x

AX = kX(1) k, x

A

11= 1

11

A

11

= 3

11

A =2 11 2

, X =

1x

AX = kX(1) k, x

A

11= 1

11

A

11

= 3

11

A =2 11 2

, X =

1x

AX = kX(1) k, x

A

11= 1

11

A

11

= 3

11

AX = kX(1) k, x

(2) (1) x x1, x2

P =1 1x1 x2

P1AP

(x1 < x2)

(3) An

AX = kX(1) k, x

AX = kX(1) k, x

AX = kX(1) k, x

,

AX = kX(1) k, x

,

AX = kX(1) k, x

,

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

AX = kX(1) k, x

,

(A kE) = 0

(2)

(2) (1)

(2) (1)

(2) (1)

(2) (1)

(2) (1)

(2) (1)

(2) (1)

(2) (1)

(3)

(3) (2)

(3) (2)

(3) (2)

n

(3) (2)

n

(3) (2)

n

(3) (2)

n

(3) (2)

n

(3) (2)

n

i)

i)

i)

ii)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

i)

(x, y)P

(x, y)P (x, y)P

(x, y)P (x, y)P

(x, y)P (x, y)P

1x

(x, y)P

o

(x, y)P

o

(x, y)P

o

=

=

(x, y)P

P(x, y)

o

=

=

(x, y)P

P(x, y)

o

=

=

(x, y)P

P(x, y)

(x, y)

o

=

=

(x, y)P

P(x, y)

(x, y)

o

=

=

(x, y)P

P(x, y)(x, y)

(x, y)

o

=

=

(x, y)P (x, y)P

1x

(x, y)P (x, y)P

1x

(x, y)P (x, y)P

1x

(x, y)P (x, y)P

1x

(x, y)P (x, y)P

1x

(x, y)P (x, y)P

1x

(x, y)P (x, y)P

1x

P

(x, y)P (x, y)P

1x

(x, y)P (x, y)P

1x

2y

(x, y)P (x, y)P

1x

2y

(x, y)P (x, y)P

1x

2y

3

(x, y)P (x, y)P

1x

2y

3

(x, y)P

o

4 y=x

(x, y)P

o

4 y=x

(x, y)P

o

4 y=x

(x, y)P

o

4 y=x

(x, y)P

o

4 y=x

(x, y)P

P

o

4 y=x

(x, y)P

P

o

4 y=x

(x, y)P

P

o

4 y=x

(x, y)P

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

(x, y)PP

o

3

4

4

4

5 x a y b

4

5 x a y b x = ax

4

5 x a y b x = axy = by

4

5 x a y b x = axy = by

xy

=a 00 b

xy

4

5 x a y b x = axy = by

xy

=a 00 b

xy

a = b = 1

4

5 x a y b x = axy = by

xy

=a 00 b

xy

a = b = 1

35 1 23 4

1(1, 0)

2

3

4

(0, 1)

(1, 2)

(3, 1)

35 1 23 4

1

1 23 4

10

=

13

(1, 0)

2

3

4

(0, 1)

(1, 2)

(3, 1)

(1, 3)

35 1 23 4

1

1 23 4

10

=

13

1 23 4

01

=2

4

(1, 0)

2

3

4

(0, 1)

(1, 2)

(3, 1)

(1, 3)

(2, 4)

35 1 23 4

1

1 23 4

10

=

13

1 23 4

01

=2

4

1 23 4

12

=3

5

(1, 0)

2

3

4

(0, 1)

(1, 2)

(3, 1)

(1, 3)

(2, 4)

(3, 5)

35 1 23 4

1

1 23 4

10

=

13

1 23 4

01

=2

4

1 23 4

12

=3

5

1 23 4

31=1

5

(1, 0)

2

3

4

(0, 1)

(1, 2)

(3, 1)

(1, 3)

(2, 4)

(3, 5)

(1, 5)

34

1

2

3

4

(2, 2)

30

45

120

150

34

1

2

3

4

(2, 2)

30cos 30 sin 30sin 30 cos 30

22

=

12

3 11

3

22

=3 1

3 1 (

3 1, 3 1)

45

120

150

34

1

2

3

4

(2, 2)

30cos 30 sin 30sin 30 cos 30

22

=

12

3 11

3

22

=3 1

3 1

cos 45 sin 45sin 45 cos 45

22

=

12

1 11 1

22

=22

0

(3 1, 3 1)

(22, 0)45

120

150

34

1

2

3

4

(2, 2)

30cos 30 sin 30sin 30 cos 30

22

=

12

3 11

3

22

=3 1

3 1

cos 45 sin 45sin 45 cos 45

22

=

12

1 11 1

22

=22

0

cos 120 sin 120sin 120 cos 120

22

=

12

1 33 1

22

=3 + 13 1

(3 1, 3 1)

(3 + 1, 3 1)

(22, 0)45

120

150

34

1

2

3

4

(2, 2)

30cos 30 sin 30sin 30 cos 30

22

=

12

3 11

3

22

=3 1

3 1

cos 45 sin 45sin 45 cos 45

22

=

12

1 11 1

22

=22

0

cos 150 sin 150sin 150 cos 150

22

=

12

3 11 3

22

=

3 13 1

cos 120 sin 120sin 120 cos 120

22

=

12

1 33 1

22

=3 + 13 1

(3 1, 3 1)

(3 1, 3 1)

(3 + 1, 3 1)

(22, 0)45

120

150

A f A

A f A

A f A

P(x1, y1) Q(x2, y2)

A f A

P(x1, y1) Q(x2, y2)f(P) = f(Q)

A f A

P(x1, y1) Q(x2, y2)f(P) = f(Q) A

x1y1

= A

x2y2

A f A

P(x1, y1) Q(x2, y2)f(P) = f(Q) A

x1y1

= A

x2y2

A

A f A

P(x1, y1) Q(x2, y2)f(P) = f(Q) A

x1y1

= A

x2y2

A1Ax1y1

= A1A

x2y2

A

A f A

P(x1, y1) Q(x2, y2)f(P) = f(Q) A

x1y1

= A

x2y2

A1Ax1y1

= A1A

x2y2

A

x1y1

=x2y2

A f A

P(x1, y1) Q(x2, y2)f(P) = f(Q) A

x1y1

= A

x2y2

A1Ax1y1

= A1A

x2y2

A

x1y1

=x2y2

PQ

A f A

A

P(x1, y1) Q(x2, y2)f(P) = f(Q) A

x1y1

= A

x2y2

A1Ax1y1

= A1A

x2y2

A

x1y1

=x2y2

PQ

36

1

2

3

2 16 3

(1, 1)

(2, 1)

(a, 3 2a)

36

1

2

3

2 16 3

(1, 1)

(2, 1)

(a, 3 2a)

36

1

2

3

2 16 3

(1, 1)

(2, 1)

(a, 3 2a)

2 16 3

11

=39

(3, 9)

36

1

2

3

2 16 3

(1, 1)

(2, 1)

(a, 3 2a)

2 16 3

21=39

(3, 9)

2 16 3

11

=39

(3, 9)

36

1

2

3

2 16 3

(1, 1)

(2, 1)

(a, 3 2a)

2 16 3

21=39

2 16 3

a

3 2a=39

(3, 9) (3, 9)

2 16 3

11

=39

(3, 9)

36

1

2

3

2 16 3

(1, 1)

(2, 1)

(a, 3 2a)

2 16 3

21=39

2 16 3

a

3 2a=39

(3, 9) (3, 9)

2 16 3

11

=39

(3, 9)

37 A

(3, 1), (4, 1)(3, 5), (5, 7) A

37 A

(3, 1), (4, 1)(3, 5), (5, 7) A

37 A

(3, 1), (4, 1)(3, 5), (5, 7) A

A

3 41 1

=3 5

5 7

37 A

(3, 1), (4, 1)(3, 5), (5, 7) A

A

3 41 1

=3 5

5 7

3 41 1

= 3 1 4 1 = 0

37 A

(3, 1), (4, 1)(3, 5), (5, 7) A

A

3 41 1

=3 5

5 7

3 41 1

= 3 1 4 1 = 0

37 A

(3, 1), (4, 1)(3, 5), (5, 7) A

A

3 41 1

=3 5

5 7

3 41 1

= 3 1 4 1 = 0

A =3 5

5 7

3 41 1

1

37 A

(3, 1), (4, 1)(3, 5), (5, 7) A

A

3 41 1

=3 5

5 7

3 41 1

= 3 1 4 1 = 0

A =3 5

5 7

3 41 1

1=3 5

5 7

1 41 3

37 A

(3, 1), (4, 1)(3, 5), (5, 7) A

A

3 41 1

=3 5

5 7

3 41 1

= 3 1 4 1 = 0

A =3 5

5 7

3 41 1

1=3 5

5 7

1 41 3

=2 32 1

P Pf

P P Pf g

P P Pf g

P P Pf g

P P Pf g

(p) f(p) g(f(p))

P P Pf gA B

(p) f(p) g(f(p))

P P Pf gA B

(p) f(p) g(f(p))Ap

P P Pf gA B

(p) f(p) g(f(p))Ap BAp

P P Pf gA B

(p) f(p) g(f(p))Ap BAp

P P Pf gA B

(p) f(p) g(f(p))Ap BAp

P P Pf gA B

(p) f(p) g(f(p))Ap BAp

P P Pf gA B

(p) f(p) g(f(p))Ap BAp

P P Pf gA B

(p) f(p) g(f(p))Ap BAp

P Pf

P P Pf gA B

(p) f(p) g(f(p))Ap BAp

P Pf

f1

P P Pf gA B

(p) f(p) g(f(p))Ap BAp

P PfA

f1

P P Pf gA B

(p) f(p) g(f(p))Ap BAp

P PfA

f1A1

38

(3, 1)

38

(3, 1)

1 23 7

131

=7 2

3 1

31

38

(3, 1)

1 23 7

131

=7 2

3 1

31

=198

38

(3, 1)

1 23 7

131

=7 2

3 1

31

=198

(19, 8)

cos sin sin cos

n=cosn sinnsinn cosn

cos sin sin cos

n=cosn sinnsinn cosn

1

cos sin sin cos

n=cosn sinnsinn cosn

1

cos sin sin cos

n=cosn sinnsinn cosn

1

cos sin sin cos

n=cosn sinnsinn cosn

1

cos sin sin cos

n=cosn sinnsinn cosn

1

?

?

?

?

?

?

?

?

?

11

11

11

?

?

?xy xy

?xy xy

?xy xy

?

?

?

?

?

?

?

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

1

1

2

1

2

1

2

2 .

1

2

2 .

1

2

2 .

1

2

2 .

1

2

2 .

1

2

2 .

11

11

11

11

11

11

11

11

11

11

11

11

11

11

11

11

(1, 0) = e1 (0, 1) = e2

(1, 0) = e1 (0, 1) = e2

(1, 0) = e1 (0, 1) = e2

(1, 0) = e1 (0, 1) = e2

(1, 0) = e1 (0, 1) = e2

(1, 0) = e1 (0, 1) = e2

(1, 0) = e1 (0, 1) = e2

(1, 0) = e1 (0, 1) = e2

(1, 0) = e1 (0, 1) = e2

(0, 1)(1, 0)

(0, 1)(1, 0)

(0, 1)(1, 0)

(0, 1)(1, 0)

(0, 1)(1, 0)

(0, 1)(1, 0)

(0, 1)(1, 0)

(0, 1)(1, 0)

(0, 1)(1, 0)

(0, 1)(1, 0)

(0, 1)(1, 0)

xy

=a bc d

xy

xy

= x

10

+ y

01

xy

=a bc d

xy

xy

= x

10

+ y

01

xy

=a bc d

xy

=a bc d

x

10

+ y

01

xy

= x

10

+ y

01

xy

=a bc d

xy

=a bc d

x

10

+ y

01

= x

a bc d

10

+ y

a bc d

01

xy

= x

10

+ y

01

xy

=a bc d

xy

=a bc d

x

10

+ y

01

= x

a bc d

10

+ y

a bc d

01

= x

ac

+ y

bd

xy

= x

10

+ y

01

xy

=a bc d

xy

=a bc d

x

10

+ y

01

= x

a bc d

10

+ y

a bc d

01

= x

ac

+ y

bd

xy

= x

10

+ y

01

xy

=a bc d

xy

=a bc d

x

10

+ y

01

= x

a bc d

10

+ y

a bc d

01

= x

ac

+ y

bd

xy

= x

10

+ y

01

xy

=a bc d

xy

=a bc d

x

10

+ y

01

= x

a bc d

10

+ y

a bc d

01

= x

ac

+ y

bd

xy

= x

10

+ y

01

xy

=a bc d

xy

=a bc d

x

10

+ y

01

= x

a bc d

10

+ y

a bc d

01

= x

ac

+ y

bd

xy

= x

10

+ y

01

xy

=a bc d

xy

=a bc d

x

10

+ y

01

= x

a bc d

10

+ y

a bc d

01

= x

ac

+ y

bd

xy

= x

10

+ y

01

xy

=a bc d

xy

=a bc d

x

10

+ y

01

= x

a bc d

10

+ y

a bc d

01

= x

ac

+ y

bd

P Pf

P Pf

(p)

P PfA

(p)

P PfA

(p) Ap

x

x

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

x y

==

==

=

=

==

=

=

==

=

=

==

=

=

==

=

=

==

=

=

==

=

=

==

=

=

==

=

=

==

=

=

==

=

=

==

=

=

==

=

=

45

5x2 + 6xy + 5y2 = 8

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

xy

=

cos 45 sin 45 sin 45 cos 45

xy

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

xy

=

cos 45 sin 45 sin 45 cos 45

xy

x = 1

2(x + y)

y = 12(x + y)

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

xy

=

cos 45 sin 45 sin 45 cos 45

xy

x = 1

2(x + y)

y = 12(x + y)

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

xy

=

cos 45 sin 45 sin 45 cos 45

xy

x = 1

2(x + y)

y = 12(x + y)

5

12(x + y)

2+ 6

12(x + y)

12(x + y)

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

xy

=

cos 45 sin 45 sin 45 cos 45

xy

x = 1

2(x + y)

y = 12(x + y)

5

12(x + y)

2+ 6

12(x + y)

12(x + y)

+5

12(x + y)

2= 8

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

xy

=

cos 45 sin 45 sin 45 cos 45

xy

x = 1

2(x + y)

y = 12(x + y)

5

12(x + y)

2+ 6

12(x + y)

12(x + y)

+5

12(x + y)

2= 8

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

xy

=

cos 45 sin 45 sin 45 cos 45

xy

x = 1

2(x + y)

y = 12(x + y)

5

12(x + y)

2+ 6

12(x + y)

12(x + y)

+5

12(x + y)

2= 8

x2

4+ y2 = 1

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

xy

=

cos 45 sin 45 sin 45 cos 45

xy

x = 1

2(x + y)

y = 12(x + y)

5

12(x + y)

2+ 6

12(x + y)

12(x + y)

+5

12(x + y)

2= 8

x2

4+ y2 = 1

45

5x2 + 6xy + 5y2 = 8

xy

=cos 45 sin 45sin 45 cos 45

xy

xy

=

cos 45 sin 45 sin 45 cos 45

xy

x = 1

2(x + y)

y = 12(x + y)

5

12(x + y)

2+ 6

12(x + y)

12(x + y)

+5

12(x + y)

2= 8

x2

4+ y2 = 1 x2

4+ y2 = 1