l l l

l
l
l

l

l

îÅ‚ Å‚Å‚
2 -3 0 T
1 2 3 1 1 1 1 2 3
ðÅ‚ ûÅ‚
· -1 4 -2 , · ,
2 -1 -1 1 0 2 0 1 2
3 -1 1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
3 -4 -5 3 29
1 1 1 1 T
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
2 -3 -3 · 2 18 , · 0 1 2 3 .
1 2 3 0
3 -5 -1 0 3
1 1 2 2 3 1
A = B =
0 2 -1 2 1 0
2A - B, AB, ABT , AT B, A3, (BT A)2, A + B - I .
îÅ‚ Å‚Å‚
3 4
ïÅ‚ śł
1 2 3 4 1 3
ïÅ‚ śł
AB BA A = , B = .
ðÅ‚ ûÅ‚
1 0 1 2 0 2
1 1
0 -1 1
B = AAT - 4I C = AT A - 4I , A = I
2 1 -2
îÅ‚ Å‚Å‚
1 0 0
ðÅ‚ ûÅ‚
0 2 0 .
0 0 3
B13 + B
"
îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 3
0 1 1 1 0 1
ïÅ‚ śł
2 2 ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
"
, 0 0 1 , 0 1 0 .
ðÅ‚ ûÅ‚
3 1
0 0 1 0 0 0
-
2 2
X l

1 0 0 0 0 1
2X - 3XT = , X + XT = , XXT = ,
5 4 0 0 1 0
0 1 3 6 1 1 -1 0
XXT = , AAT X = , A = .
1 1 1 2 0 2 1 -2
1 0 1 1 2 1 2 1 29 12
· X = , · X · = ,
2 1 2 1 2 0 3 1 14 6
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
0 2 1 2
0 0 0
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
XT · 1 2 3 = , X + 0 2 = 2X - 1 1 ,
1 2 3
2 1 0 1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 1 1 0 5 0 0 0 3
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
1 2 2 · X + -5 0 -3 = 2 0 8 .
1 2 2 -4 -3 -3 4 5 5
X Y XA = I + Y
l
Y l I

1 -1 1
A = .
0 2 3
X l

1 1 4 0 -1 1
X2 = , X2 = , X2 = .
0 1 1 1 0 -1
A3 = 0
l
n
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 0 1 cos x sin x 0
1 1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A = , B = 0 1 0 , C = - sin x cos x 0 .
0 1
1 0 1 0 0 1
A l A = -AT

l

B = [bij] X X b11 = 3 b12 = 1 b31 =
2 1 -1
-2. (AX)T = B + AT , A =
0 1 1
A B = A + AT
C = A - AT
îÅ‚ Å‚Å‚
1 2 3
ðÅ‚ ûÅ‚
4 5 0
2 1 1
A 2 l AAT = I

1 1 1
AX = I A = X = A-1 XA
0 1 1
+
. A-1 l

[A, I]
[I, X] X A
[I, X]
l
A-1
îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 2 3 1
1 -2 4 0 -1 1 -5 3 1
ïÅ‚ śł
1 0 1 1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ïÅ‚ śł
A = 0 1 -2 , B = -1 2 -1 , C = 2 -4 -1 , D = .
ðÅ‚ ûÅ‚
3 1 4 1
0 0 1 2 -1 0 0 5 1
0 1 1 2
1
C-1DT
2
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 -1 1 0 2 0 0 -1
ïÅ‚ śł ïÅ‚ śł
1 1 -1 0 1 18 18 17
ïÅ‚ śł ïÅ‚ śł
C = , D = .
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
-1 1 1 0 2 0 0 1
0 0 0 1 4 0 0 2
-1
2 1 2 1 1 4
· X · = ;
0 1 0 1 2 2
-1
3 4 3 4 0 1
· X · = ;
1 1 1 1 0 0
îÅ‚ Å‚Å‚-1 îÅ‚ Å‚Å‚-1 îÅ‚ Å‚Å‚
1 -4 -3 1 -4 -3 5 16 17
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
1 -5 -3 · X · 1 -5 -3 = -2 3 1 .
-1 6 -4 -1 6 -4 4 0 3
1 2 3 1 0 2 1 1 1
4 5 0 , 0 3 0 , 1 2 3 .
6 0 0 2 0 5 2 5 6
1 1 1
sin x cos x
1 0 ex
=
- cos x sin x
1 e-x 0
x
0 4 0 2 0 1 2 6 1 3 2 0
0 1 2 3 0 0 8 7 5 0 1 2 0 6 0 2 0 1 3 0
1 2 3 0 0 0 6 5 4 3 2 1 1 0 2 0 1 0 1 1
, , , .
2 3 0 0 4 3 0 0 2 0 1 1 0 0 2 2 0 0 4 6
3 0 0 1 2 1 0 0 0 2 0 0 3 0 0 0 5 1 1 1
0 1 0 0 6 0 5 9 2 4 8 4
C = AB D = ABT
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 2 3 4 1 0 0 0
" "
ïÅ‚ śł ïÅ‚ śł
0 1 2
ïÅ‚ śł ïÅ‚ " śł
"3 , B = ðÅ‚ 2 1 0 0 .
A =
ðÅ‚ ûÅ‚ ûÅ‚
5 5 5 0
0 0 1 7
"
0 0 0 5 7 8 8 4
l

1 1 1 1 1 0 1 1 1 1
-1 0 1 1 1 1 2 3 3 3
-1 -1 0 1 1 , 1 3 2 3 3 .
-1 -1 -1 0 1 1 3 3 2 3
-1 -1 -1 -1 0 1 3 3 3 2
l l

-1 2 3 4 1 0 1 1 1 1
1 -1 3 4 2 1 0 1 1 1
1 2 -1 4 3 , 1 1 0 1 1 .
1 2 3 -1 4 1 1 1 0 1
1 2 3 4 -1 1 1 1 1 0
A = [ aij ] 6 aij
x dla i j
aij = .
y dla i > j
A B C det A = 128
det B = 4 det C = 2 .
det (2BCT ) , det ((A-1B)T (2C))-1
A A = -AT
A n
1798 2139 3255 4867 31
1 7 9 8
2 1 3 9
3 2 5 5
4 8 6 7
31
0 1
l

A B = S-1AS SB = AS
detA = detB
X X2 - XT = 0 .
l
l

a1 + b1 a2 + b2 a3 + b3 a1 a2 a3
b1 + c1 b2 + c2 b3 + c3 = 2 b1 b2 b3 ,
a1 + c1 a2 + c2 a3 + c3 c1 c2 c3
a1 + b1x a2 + b2x a3 + b3x a1 a2 a3
a1 - b1x a2 - b2x a3 - b3x = -2x b1 b2 b3 .
c1 c2 c3 c1 c2 c3
l

a2 b2 c2 d2
1 1 1
(a + 1)2 (b + 1)2 (c + 1)2 (d + 1)2
a b c ,
.
(a + 2)2 (b + 2)2 (c + 2)2 (d + 2)2
b + c a + c a + b
(a + 3)2 (b + 3)2 (c + 3)2 (d + 3)2
l x, y " R l

îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚
0 x 0 x
x 0 y
ïÅ‚ śł
cos x ex x 0 x 1
ðÅ‚ ûÅ‚ ïÅ‚ śł
, 0 1 0 , .
ðÅ‚ ûÅ‚
e-x cos x 0 x 1 x
y 0 x
x 1 x 1
A AX = B Y A = B
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
2 1 0 1 7 1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A = 1 0 -1 , B = 2 3 0 .
1 2 2 -2 5 1
l l

îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
3 2 1 2 3 3 -4 -3
32 14 -1 2 0 2
ïÅ‚ śł ïÅ‚ śł
7 5 2 5 0 6 1 1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ïÅ‚ śł ïÅ‚ śł
2 1 0 , 3 1 4 , , .
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
0 0 9 4 5 4 2 1
25 11 -1 4 2 6
0 0 11 5 2 3 3 2
+
x " R A = [aij] n 4
Å„Å‚
x dla i = j
òÅ‚
i dla i = j < n
aij = , aij = j - 1 dla i < j .
x dla pozostalych
ół
j dla i > j
+
A = [ aij ] n 4
0 dla i = j 2
aij = , aij = i · j2,
1 dla pozostalych
Å„Å‚
Å„Å‚
i dla i = j
ôÅ‚
ôÅ‚
1 dla |i - j| = 1
òÅ‚ òÅ‚
j dla i = 1
aij = 2 dla i = j , aij =
ół ôÅ‚ -i dla j = 1, i 2
ôÅ‚
0 dla pozostalych
ół
0 dla pozostalych
+
Un, Wn, Vn n 2
5 3 0 . . . 0 0
2 5 3 . . . 0 0
0 2 5 . . . 0 0
Un = = 3n+1 - 2n+1
0 0 0 . . . 5 3
0 0 0 . . . 2 5
1 1 1 . . . 1 1
1 2 2 . . . 2 2
1 2 3 . . . 3 3
Wn = = 1
1 2 3 . . . n - 1 n - 1
1 2 3 . . . n - 1 n
a -b 0 . . . 0 0
0 a -b . . . 0 0
0 0 a . . . 0 0
Vn = = an - bn
0 0 0 . . . a -b
-b 0 0 . . . 0 a
l

l

Å„Å‚ Å„Å‚
2x
òÅ‚ - 3y - 4z = -6 4x + 3y + z = 8
òÅ‚
-x + y + z = -2 2x - 2y - 3z = -3
ół ół
3x + y + 5z = 2 -2x + 15y + 16z = 29
Å„Å‚ Å„Å‚
x
ôÅ‚ - y - z = 1 15x1 + 12x3 - 3x3 - x4 = 14
ôÅ‚
ôÅ‚ ôÅ‚
òÅ‚ òÅ‚
3x + 4y - 2z = -1 7x1 + 12x2 + 4x3 + x4 = 8
3x
ôÅ‚ - 2y - 2z = 1 12x1 - 3x3 + 2x4 = 14
ôÅ‚
ôÅ‚ ôÅ‚
ół ół
x - 3y + 3z = -1 - 10x2 + x3 + 4x4 = 4
Å„Å‚
Å„Å‚
x + 5y + z = 0
ôÅ‚
ôÅ‚
x1 + 2x2 - 3x3 - 3x4 = 4
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
òÅ‚ -x + y + 7z = 2
òÅ‚
2x1 - x2 - 3x3 - x4 = 3
3x + 7y - z = 4
3x1 + x2 - 4x3 - 4x4 = 7
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
x + 3y + 3z = 4
ôÅ‚ ół
ôÅ‚
x1 - 3x2 + 2x3 + 2x4 = 1
ół
2x + 5y + 2z = 5
Å„Å‚
Å„Å‚
x + y - z - 2u = 5
ôÅ‚
ôÅ‚
x + 2y = 3
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
x
òÅ‚ - y - z = 1
òÅ‚
2x - y = 0
2x - 2z - 2u = 6
4x + 3y = 1
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
3x
ôÅ‚ - y - 2z - 2u = 7
ół
ôÅ‚
5x - y = 4
ół
2x - 2y + z = 2
x + 2y + 3z = 1
l l

x - 2y + 2z = 2
l

p " R
l
Å„Å‚
2px + 4y - pz = 1
òÅ‚
(p - 2)x + py = 1
2x + y + pz = 2
-3x + (p + 2)y = p
ół
(4 + 2p)x + 6y + pz = 3
Å„Å‚
Å„Å‚
x
ôÅ‚ - y - z - t = px
ôÅ‚
px + 3y + pz = p
òÅ‚ òÅ‚
-x + y - z - t = py
px - 2z = 1
ôÅ‚ -x - y + z - t = pz
ół
ôÅ‚
x + 2y + pz = p
ół
-x - y - z + t = pt
l

l

Å„Å‚ Å„Å‚
x + 2y + 3z = 6 x + 2y + 3z = 14
òÅ‚ òÅ‚
2x + 3y + z = 6 3x + y - z = 2
ół ół
3x + y + 2z = 6 5x + 7y + 8z = 43
y l

Å„Å‚
Å„Å‚
x
ôÅ‚ - 2y + 3s + t = 1
ôÅ‚
x + y + z + t = 1
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
2x
òÅ‚ òÅ‚ - 3y + z + 8s + 2t = 3
2x + 2y + z + t = 0
x - 2y + z + 3s - t = 1
3x + 2y + 3z + 2t = 3
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
y + 3s + 5t = 0
ół ôÅ‚
ôÅ‚
6x + 4y + 3z + 2t = 2
ół
x - 2y + 5s + 8t = -1
A AX = I
A
A n l

l A B1, B2

AX1 = B1, AX2 = B2.
l

AX = B X = [X1, X2] B = [B1, B2]
AX = B
" l l A B

l l l l

l l

l l

" l l

l

l

l

l B

l AX1 = B1

AX2 = B2
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 1 1 6 3
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A = 2 1 -1 , B1 = 1 , B2 = 0 ,
3 2 -1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚4 îÅ‚ Å‚Å‚1
1 2 1 1 1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A = 3 4 1 , B1 = 1 , B2 = 2 .
0 1 1 1 3
A-1
AX = I
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 2 5 2 1 1
1 2
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A = , B = 0 1 2 , C = 0 1 1 .
1 1
0 0 1 1 0 1
+
A n A-1
AX = I. n
l

AXi = Ei i = 1, . . . , n,
Ei I n Xi
X = A-1 A-1
l C = [A, I]

[I, X] X A-1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 0 0 1 3 2 0 0
32 14 -1 1 1 1
ïÅ‚ śł ïÅ‚ śł
0 0 1 1 7 5 0 0
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ïÅ‚ śł ïÅ‚ śł
2 1 0 , 1 2 2 , , .
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
0 1 1 1 0 0 9 4
25 11 -1 1 1 2
1 1 1 1 0 0 2 1
îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚
3 2 -1 2 0 1
îÅ‚ Å‚Å‚
1 3 5 -1
ïÅ‚ śł
2 1 3 -2 4 4 1 0 -3 0 2
ïÅ‚ śł ïÅ‚ śł
2 -1 -3 4
ðÅ‚ ûÅ‚ ïÅ‚ śł ïÅ‚ śł
4 2 5 -2 7 , , 2 -1 -2 1 1 -3 .
ðÅ‚ ûÅ‚ ïÅ‚ śł
5 1 -1 7
ðÅ‚ ûÅ‚
2 1 1 8 2 3 1 3 -9 -1 6
7 7 9 1
3 -1 -5 7 2 -7
p
îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
p 1 1 1
îÅ‚ Å‚Å‚
3 1 1 4 1 2 -1 1
ïÅ‚ śł
1 p -1 2 1 p 1 1
ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
p 4 10 1 5 1 2 1
ðÅ‚ ûÅ‚ ïÅ‚ śł ïÅ‚ śł ïÅ‚ śł
2 -1 p 5 , , , 1 1 p 1 .
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ïÅ‚ śł
1 7 17 3 4 -1 p 0
ðÅ‚ ûÅ‚
1 10 -6 1 1 1 1 p
2 2 4 3 3 p 4 -1
1 1 1 1
x = 2 y = 3 z = 4 l

Å„Å‚ Å„Å‚
x
òÅ‚ - y - 2z = -9 x + 2y - z = 4
òÅ‚
px + y + z = 9 x - py + 2z = 4
ół ół
2x + py = 7p +4x + 4y - z = 8p
 l

p " R l

Å„Å‚ Å„Å‚
2x + 3y + z + 2t = 3 2x
ôÅ‚ ôÅ‚ - y + z + t = 1
ôÅ‚ ôÅ‚
òÅ‚ òÅ‚
4x + 6y + 3z + 4t = 5 x + 2y - z + 4t = 2
6x + 9y + 5z + 6t = 7 x + 7y - 4z + 11t = p
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ół ół
8x + 12y + 7z + pt = 9 3x + 6y - 3z + 12t = p + 1

l
Å„Å‚ Å„Å‚
Å„Å‚
x + 2y + 3z = -1 2x
ôÅ‚ ôÅ‚ - 2y + 6z = -7
ôÅ‚ ôÅ‚
2x + y - z + 4u = 1
òÅ‚ òÅ‚ òÅ‚
2x + 4y + 5z = 2 6x + y + 8z = 5
x - y + 2z - 3u = 2
3x + 6y + 7z = 5 4x + 3y + 2z = 12
ół ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
3x + z u = 3
ół ół
4x + 8y + 11z = 0 4x + 5y + 2z = 1
l p

Å„Å‚
Å„Å‚ Å„Å‚
2x
ôÅ‚ - y - 3z + 4u = 5
ôÅ‚
px + y + z = 1 px + y + z = 1
òÅ‚ òÅ‚ òÅ‚
4x - 2y + 5z + 6u = 7
x + py + z = 1 x + py + z = p - 1
6x
ôÅ‚ - 3y + 7z + 8u = 9
ół ół
ôÅ‚
x + y + pz = 1 4x + y + pz = (p - 1)2
ół
px - 4y + 9z + 10u = 11
l p

Å„Å‚ Å„Å‚
2x + y + z = 2 px + py + (p + 1)z = p
òÅ‚ òÅ‚
x + 3y + z = 3 px + py + (p - 1)z = p
ół ół
2x + y + pz = p (p + 1)x + py + (2p + 3)z = 1
Å„Å‚ Å„Å‚
(p + 3)x + y + 2z = p x + py = 1
òÅ‚ òÅ‚
px + (p - 1)y + z = 2p py + z = 0
ół ół
3(p + 1)x + py + (p + 3)z = 5 px + z = -1
l

2 + x + 2y - z - t = 1 + x + y + z + 3t = 3x + 5y - z + t = 3
2x - y + z + 3t = 8x + 6y + 10z + 14t = 5 + x + 2y + 2z + 2t = 4
x1 + x2 + 5x3 - 2x5 + 3x6 + 1 = 4x1 - 3x3 + x4 + x5 + 3x6 + 1 = x1 + 6x3 + x5 + x6 = 2
35 l 94

l l 1, 8 dm2

2, 1 dm2
l 200

l

l l 135 135

210 235
A(-1, 9) B(1, -1) C(2, -3), A(1, 1) B(2, 3) C(3, 5)
 l
l
l

l l l 350 690 170

200 205
l

x, y " R l

" "
x(3 - 2i) + y(4 - 5i) = 10 - 9i x(- 2 + i) + y(3 2 + 5i) = 8i
x(4 - 3i)2 + y(1 + i)2 = 7 - 12i (2 + 3yi)(x - 2i) = 2 + xi
2
x y 2 + i 4 - i
+ = 1 x + y = 1 + i
3 + i 1 - 3i 3 - i 1 - 3i
l

2z + z + 5i = 6 2z + (1 + i)z + 3i = 1 4z = z2 + 4
z, u l l

(2 + i)z + (2 - i)u = 6, (4 + 2i)z - (2 + 3i)u = 5 + 4i,
(3 + 2i)z + (3 - 2i)u = 8, (3 - i)z + (4 + 2i)u = 2 + 6i.
z z2

l
z2 = z , z3 = z , (z)2 = z2
z1 = 0 z2 = 3 + 2i z3 = 2 + 3i l l

l l

l l

Re (iz - 1) 0 Im (z2) > 0 z + i = z - i
1 - z
9 z · z 0 < |z + i| < 4 Re = 1
1 + z
" "
3 + 3i , 3 3 - 3i , -5 + 5 3 i
z z
25
"
Ä„ Ä„
(2 - 2i)10 (2 3 - 2i)30 cos - i sin
5
" " "5 15
(1 + i)22 ( 3 + i)(-1 - 3 i) - 3 + i
"
1 - i 2
(1 - 3 i)6
12 12
Ä„ Ä„
1 + i ctg + 1 - i ctg = 0
24 24
z = (1 + i) + (1 + i)2 + (1 + i)3 + . . . + (1 + i)8.
 l l a

l

Ä„ Ä„
Ä„ arg (iz) < 2Ä„ arg (-z)
3 2
Ä„
|z - 1 - 2i| > 3 |z - 3| < 5 arg (-z) >
2
Ä„
|z + 4| 6 Im z > 0 arg z > |z + 2i| < 2 -Ä„ < arg z < Ä„
2
z l |z|2 + |iz|2 = |z - iz|2

"
z = -1 + 3 i l l l

l z0 = 1, z0 = 3 + i
l

l
" " "
3
4
-11 + 60i , i , -4

l
"
4
3 4
(5 - 4i)4 (2 - 2i)9 (-2 + 3i)4 ( 3 - i)12
z · z4 = -32, z · z3 + z · 8i = 0,
(i + z)4 = (-1 - z)4, (i - z)4 = (z - 1)4
z n w
z w
n n

l
"
8i , 2 - 2i , (- 3 + i)3 , (1 + i)20
l

"
1 3
z3 = 8i, z4 = -4 , z4 = - - i
2 2
. l

|z4| = z , z3 · (z)2 = -1 , z3 = (2 + 2i)6.
. l

1
z7 = z (z5) = z2|z3| (z)2 |z4| =
z2
|z|4 = iz4 z6 = (z)6 |z4| = z2
. l cosÕ

2
. n r r 2 - 2 cosÕ
l
2Ä„
Õ = l

n
l

 P Q
P (x) = x5 + x2 + x + 1, Q(x) = x2 - 1
P (x) = x7 - x5 + x4 + x3 + x + 3, Q(x) = x3 - x
P (x) = 2x5 + 3x4 + 2x3 + 3x2 + 3x + 2, Q(x) = x2 + 1
P (x) = x5 - 2x4 + 3x3 - 4x2 + 5x, Q(x) = x2 - 2x + 2
l

W (x) = x4 - 5x3 + 7x2 - 5x + 6, x1 = i
W (x) = x4 - 5x3 + 10x2 - 10x + 4, x1 = 1 - i
W (x) = x5 - x4 + 4x3 + 4x2 + 3x + 5, x1 = 1 + 2i
W (x) = x5 + 8x4 + 22x3 - 18x2 - 19x + 30, x1 = 2 - i
l

2 - 2i, 2i 3
2 - i, 1 - i, i -1
1 + 2i, -3 1 + i
l

P (x) = x5 - 2x4 + 3x3 - 4x2 + 6x - 4 P (x) = x6 - 1
P (x) = x5 - 2x2 - x + 2 P (x) = x4 - 81
P (x) = x7 + x6 + x5 + x4 + x3 + x2 + x + 1
l l

2x2 - 2x + 3 2x3 + 3x2 + 4x - 3 5x - 12 x3 - 8x2 - 14x - 13
(x2 - 2x + 2)(x2 + 1) (x2 - 1)(x2 + 2) x2 - 5x + 6 x4 - x3 - 5x2 - x - 6
A(3, -1, 2) B(1, 2, -4) C(-1, 1, 2) l l ABCD.

l D

A(3, -1, 2) B(1, 2, -1) C(-1, 1, -3) D(3, -5, 3) l l

B(2, 0, 2) C(5, -2, 0) AD
l
A D
l u = [ 1, 1, 2 ] v = [ 3, -1, 2 ] u = [ 0, 2, 3 ]

p " R u = [ 0, 1, 1 ] w = [ p, 2, p ] l

Ä„
p " R u = [ 0, 1, 1 ] w = [ p, 4, p ]
3
A(2, 4, 6) B(0, 0, 2) C(0, p, p) l

l B. p.

u + v u - v l u v l

u + v u - v l u v l

u w
|u - w|2 = |u|2 + |w|2 - 2 |u| · |w| · cos Õ ,
Õ u w.
 u = -2 p + 4 q v = 3 p + q,
p q 600 |p| = 3 |q| = 2.
p " R l A(2, 3, 2) B(3, 5, 7)

"
C(p, 0, p) 4 3
a, b " R A(0, 2, 1) B(1, 2, 3) C(a, b, 7)
AC ABCD l a P BP

P E E AD l l AP

l u v


l
l

p " R ABCD l A(1, 0, 1)

B(1, 1, 2) C(2, 1, 1) D(p, p, p) 1
p " R A(1, 1, 1) B(1, 1, 2) C(p, 1, 0) D(2, 0, p)

l
P (x, x, 2) x " R
l
A(0, 1, 2) B(1, 2, 3) C(4, 1, 3)
l P (1, -2, 5) l A(0, -5, 1) B(6, 3, 2)

C(-3, -9, 1)
l z l z = 1

A(0, 3, 0) B(1, 2, 2)
+
l A(3, 0, 0) B(0, 1, 0)

l x y
l
A(-1, 1, 0) B(1, 5, -4)
P (2, -5, 0) l l

x - 3z + 5 = 0
P (7, 2, 0)
l
u = [ 3, -2, -3 ], v = [ 1, 2, -3 ]
l Ä„1

A(1, 7, 8) B(2, 8, 8) C(-4, 2, 7) l Ä„2 : x + 2z - 4 = 0

Å„Å‚ Å„Å‚
x = 1 + t x =
òÅ‚ òÅ‚ -1 - 2s
l1 : y = 2 + t , l2 : y = 3 + s, s, t " R
ół ół
z = 4 + 2t z = -4 - 8s
Å„Å‚
x = 1 + t
òÅ‚
l Ä„ : x + y - z + 3 = 0 l : y = 3 + 2t , t " R

ół
z = 5 + 3t
Å„Å‚ Å„Å‚
x = 1 + t x =
òÅ‚ òÅ‚ -1 + Ä… + ²
+
. l : y = 2 - t , l Ä„ : y = 2 + 3Ä… - ² , t, Ä…, ² " R

ół ół
z = 3 - 2t z = 3 + 2Ä… + 2²
l l

Ä„1 : x - y + z = 0 Ä„2 : 5x + y - z + 24 = 0
Å„Å‚ Å„Å‚
x =
òÅ‚ -1 + 2t x =
òÅ‚ -1 - 4s
l1 : y = -1 - t , l2 : y = -1 + 4s , s, t " R
ół ół
z = 2 + 2t z = 2 + 2s
Å„Å‚
x = 1 + 2t
òÅ‚
l M(6, 6, 3) l1 : y = -1 - t , t " R

ół
z = 3 - 2t
Å„Å‚
x = 1 + t
òÅ‚
x + y - z + 2 = 0
l1 : l2 : y = -3 + 2t , t " R
x - 4y + 3 = 0
ół
z = 2 + 3t
Å„Å‚
x = 1 + 3s
òÅ‚
5x - y - z + 1 = 0
l3 : l4 : y = -2 - s , s " R
3x - z + 4 = 0
ół
z = 2s
l l Ä„

Å„Å‚
x = 3 + s
òÅ‚
l : y = 2 + 2s , s " R, Ä„ : 4x + y + 5z - 13 = 0
ół
Å„Å‚z = 4 + 3s
x = 7 + 3t
òÅ‚
l2 : y = 1 - t , t " R, Ä„ : x + 2y + 3z + 3 = 0
ół
z = 2 + 2t
+
. M(1, 2, 1) l

Ä„1 : 6x - 3y + 6z + 3 = 0, Ä„2 : 4x - 2y + 4z - 2 = 0.
+
. (-2, 4, 3) B(1, -2, 2) l Ä„ : 2x + 3z - 7 = 0

A(3, 1) B(-1, 1) C(3, -3)
A(3, 1) B(-1, 1) y = x
A(-1, -2) l
l
+
. (x -x0)2 +(y - y0)2 = r2 P (x1, x2)
l P (-4, -4)

l l l

l M(-1, 1) N(1, 3)

l

A(4, 0) B(0, -3) l


l
l 9x2 - 18x + 4y2 + 16y - 11 = 0
l
P (-1, 0) F1(1, 2)
F2(5, 2)
P (6, 1) F1(-3, 1)
F2(7, 1)
l l x2 - 6x - 9y2 - 54y = 153

 l l x2 - y2 = 1

l l xy = 1

l

y2 - 4y - 4x + 8 = 0, x2 - 6x - 8y + 49 = 0
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 3 5 1 0
9 2 -1 6
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
; 1 2 3 ; 0 1 ; .
2 -9 1 8
1 4 7 -1 -6
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
2 3 1 24 44 38
0 -1 3 7 3
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
; ; 6 5 1 ; 26 48 41 l

-2 3 -2 5 2
2 5 2 7 13 11
îÅ‚ Å‚Å‚
7 6 13 20
ïÅ‚ śł
9 20 4 2 6 10
ïÅ‚ śł
AB = BA = .
ðÅ‚ ûÅ‚
5 8 2 0 2 4
2 2 4 6Å‚Å‚
îÅ‚
0 2 -4
-2 -3
ðÅ‚ ûÅ‚
B = ; C = 2 -2 -3 .
-3 5
-4 -3 1
îÅ‚ Å‚Å‚
x 0 0
ðÅ‚ ûÅ‚
0 y 0 , x, y, z
0 0 z
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
"
0 1 2 2 0 2
1 3
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
" ; 0 0 2 ; 0 2 0 .
- 3 1
0 0 2 0 0 0
-1 -3 0 x
; , x
-2 -4 -x 0
1 2
.
0 0
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 - x 1 4
2 1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
-1 , x ; 0 1 ; 1 3
3 3
x 2 2
îÅ‚ Å‚Å‚
T 0 0 5
1 1
2 0 0
ðÅ‚ ûÅ‚
X = , Y = 0 0 3 .
1 1 0
2 2
0 0 -2
1 1
1 -1 - 2 0 2 0 -2 0 -2 0
2 2
, ; , , , ;
1 1
0 1 0 -1 1 1 -1 -1 1 - -1
3 3
0 0
x
x 0
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
2n-1 0 2n-1 cos nx sin nx 0
1 n
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
An = ; Bn = 0 1 0 ; Cn = - sin nx cos nx 0 .
0 1
n-1
2n-1 0 0 0 1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚2
0 2 -1 3 1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
X = -2 0 4 , B = -8 4 .
1 0 -2 -4
îÅ‚ Å‚Å‚-4îÅ‚ Å‚Å‚
1 3 2 0 -1 1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
3 4 0 + 1 0 0 .
2 0 2 -1 0 0
1 0 1 0 -1 0 -1 0
, , ,
0 1 0 -1 0 1 0 -1
îÅ‚ Å‚Å‚
1 -1
ðÅ‚ ûÅ‚
X = x y x, y
-x 1 - y
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 2 0 1 1 1 -1 -2 -1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A-1 = 0 1 2 ; B-1 = 2 2 1 ; C-1 = 2 5 3 ; D-1
0 0 1 3 2 1 -10 -25 -14
îÅ‚ Å‚Å‚
1 9 9 8
ïÅ‚ śł
1 9 9 9
ïÅ‚ śł
.
ðÅ‚ ûÅ‚
2 0 0 0
2 0 0 1
îÅ‚ Å‚Å‚
1 0 1
1
-1 3 -9
2 ðÅ‚ ûÅ‚
; ; 1 1 0 .
4 4 1 -3
0 0 1
-90; 3; -2
x = 0
85; -4; -27; 0
detC = detD = detA · detB = 100
1; 4
1080; 4
x(x - y)5
128, 1
n
detS = 0

det A = 0 det A = 1
x = kĄ, k " Z; x2 = y2; x = 0

îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
0 3 1 -17 23 12
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
X = 1 1 -1 , Y = -5 8 4 .
-2 0 1 -17 21 11
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚
5 -2 -5 4 -7 5 12 -19
1 -3 -1
ïÅ‚ śł ïÅ‚ śł
-7 3 16 -13 3 -2 -5 8
ðÅ‚ ûÅ‚ ïÅ‚ śł ïÅ‚ śł
-2 7 2 ; ; .
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
0 0 5 -4 41 -30 -69 111
3 2 -4
0 0 -11 9 -59 43 99 -159
n(1-n)
0, 1, 2, ..., n - 1; 1, 2, 3, ..., n - 1,
2
n(n+1)
(-1)n 0; n + 1; n!
2
l

l x = 1 - z, y = 2 - z, z l

x1 = 1 - x2, x3 = -2x2, x4 = 1 + 3x2 x2 x = 4 y = -1 z = 1
x1 = 2 + x3 + x4, x2 = 1 + x3 + x4, x3, x4 x = 3 + z + u, y = 2 + u, u, z
l

p = -4, p = 1; p = -2, p = 2 p " R.

x = y = z = 1; x = 1, y = 2, z = 3.
y = -2; y = 3
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 1 -1 + t
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
X1 = 2 X2 = 0 X1 = 1 - t t l

3 2 t
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 -2 -1 1/2 -1/2 0
-1 2
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A-1 = ; B-1 = 0 1 -2 ; C-1 = 1/2 1/2 -1 .
1 -1
0 0 1 -1/2 1/2 1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
0 0 -1 1 5 -2 0 0
1 -3 -1 2 -1 0
ïÅ‚ śł ïÅ‚ śł
0 -1 1 0 -7 3 0 0
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ïÅ‚ śł ïÅ‚ śł
-2 7 2 ; 0 1 -1 ; ; .
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
-1 1 -1 1 0 0 1 -4
-2 0 1 -1 0 1
1 0 1 -1 0 0 -2 9
p = 3 p = 3 p = 0 p = 0 p = -3 p = 3 l

p p = 1 p = 1

p " R; p = 5.

l

p = 8 p = 8

(p - 1)(p + 2) = 0 l p = 1

p = -2 l

(p - 1)(p + 2) = 0 l (p - 1)(p + 2) = 0 l

p + 1 3p - 2 p - 2
p = 1 l p = 1 x = y = z =

5p - 5 5p - 5 p - 1
p = 0, x = 1 - p, y = p, z = 0, p = 0 x = 1, z = 0, y

p2 + 4p - 15 p2 + p + 15 -4p2 + p + 15
p(p - 1) = 0 x = y = z =

p2 p2 p2
p = 1 x = 2 - z y = -7 + 2z z p = 0 l

1
p(p + 1) = 0 x = 0 y = z = -1, p = -1 x = 1 + z y = z z p = 0

p
l

l x = 11 + 8y + 4z, t = -6 - 5y + 3z, y, z

y = 5 - 3x - 17z - 5u, s = -1 - 3x + 9z - 2u, t = 2 - x - 6z - u, x, z, u
23 12
25, 2 m2 1200
1 2 3 4 l

y = x2 - 5x + 3
l l

x = 2 y = 1 x = 3 y = 1 x = 1 y = 6 x = 3 y = 1 x = 2 y = 0
2 - 5i 2 - 5i 2 -2 + 4i -2 - 4i
z = 2 + i u = 2 - i z = 1 + i u = i
z1 = x + 0i z2" 0 + yi x, " R
=
"y
1 3 1 3
0, 1, - + , - - 0, 1, -1, i, -i z = x z = yi x, y " R
2 2 2 2
z4 = -1 + i
l l Im z -1

l l
l
3 l
l
l 4 z = -i

1
-1 z = -1
2 2
"
Ä„ Ä„ 11Ä„ 11Ä„ 2Ä„ 2Ä„
3 2 cos + i sin 6 cos + i sin 10 cos + i sin
4 4 6 6 3 3
z = |z| (cos(-Õ) + i sin(-Õ))
-215i -430 -1 -32i 2 - 2i i
15 - 15i
Ä„
2
l Im z

l

 l Im z l l

"
z = -1 - 3 i l Im z

l

z0 = 3 5
l l
3 w0 = -1 - 2i
l l
l
l 6 z0 = -4 l l

l l

l l 2 z0 = -2i l l

" " " " " "
- 3 - i, 1 - 3 i, 3 + i 1 - 3 - 2i, 3 - 3 i, 1 + 3 + 2i
" " "
4 - 3 - 3i, 7 + 2i - 3 i, 2 + 3 + 5i
" "
3 1 3 1
{5 + 6i, -5 - 6i} -i, + i, - + i { 1 + i, -1 + i, -1 - i, 1 - i }
2 2 2 2
" " " "
{9 - 40i, -9 + 40i} -16(1 + i), 8 1 + 3 + (1 - 3)i , 8 1 - 3 + (1 + 3)i
{-2 + 3i, -3 - 2i, 2 - 3i, 3 + 2i} {8, 8i, -8, -8i}
" " " "
1 1 1 1
1 + 3 i -2 1 - 3 i 2i - 3 - i 3 - i 0, -1 - i, - - i 0, 1 + i, + i
2 2 2 2
0 1 n -1 n
"
Ä„ 7Ä„ Ä„
2 4 2
8ei 2 2ei 8ei 210eiĄ
" " " "
" "
1 3 3 1 1 3 3 1
-2i, - 3 + i, 3 + i 1 + 1, 1 + i, -1 - i, 1 - i + i, - + i, - - i, - i
2 2 2 2 2 2 2 2
" "
0, 1 -1 8i, 4 - 4i, -4 3 - 4i
" " "3 " " " " "
2 2 2 2 2 2 2 2
0, 1, + i, i, - + i, -1, - - i, -i, - i
2 2 2 2 2 2 2 2
l
l
l 1

l l l

3Ä„
8
l
l
Ä„ Ä„
6 3
-1, 0, 1.

l
R(x) = 2x + 2 R(x) = x2 + 2x + 3 R(x) = 3x + 2 R(x) = -x + 4
-i, 2, 3 1 + i, 1, 2 1 - 2i, -i, i, -1 2 + i, -1, 2, 3
(x2 - 4x + 8)(x2 + 4)(x - 3)3 (x2 - 4x + 5)(x2 - 2x + 2)(x2 + 1)(x + 1)2
(x2 - 2x + 5)(x + 3)(x2 - 2x + 2)2
(x2 - 2x + 2)(x2 + x + 2)(x - 1) (x2 - x + 1)(x2 + x + 1)(x - 1)(x + 1) (x2 + x + 2)(x - 1)2(x + 1)
(x2 + 9)(x + 3)(x - 3) (x2 + 2x + 2)(x2 - 2x + 2)(x2 + 1)(x - 1)(x + 1)
1 1 1 1 3 2 3 1 -2 2x + 1
+ + + + + +
x2 + 1 x2 - 2x + 2 x - 1 x + 1 x2 + 2 x - 2 x - 3 x + 2 x - 3 x2 + 1
D(1, -2, 8)
A(-1, 2, 4) B(8, -4, -2)
[ 2, 0, 3 ] [ 4, -2, 1 ] [ 4, 2, 7 ] [ -2, 4, 3 ]
p = -2
 p = -1
p = 1
-8
p = 0 p = 4
a = 3, b = 2
"
3
0 2a
4
p = -4 p = 8
p = 1
3 3
P ( , , 2)
4 4
1.
5x - y - 3z + 3 = 0
Å„Å‚
x = 3s
òÅ‚
x + 3y - 3 = 0 y = 1 - s , s, t " R
ół
z = t
Å„Å‚
x =
òÅ‚ -1 + t
y = 1 + 2t , t " R
ół
z = -2t
Å„Å‚
x = 2 + t
òÅ‚
y = -5 , t " R
ół
z = -3t
Å„Å‚
x = 7 + 6t
òÅ‚
y = 2 + 3t , t " R
ół
z = 4t
Å„Å‚
x = 4
òÅ‚ - t
y = 10 - 2t , t " R
ół
z = t
P (1, 2, 4)
P (0, 2, 5)
l
Å„Å‚ Ä„1 : x + 2y - 2z + 12 = 0 Ä„2 : 4x - y + z - 12 = 0
Å„Å‚
x =
òÅ‚ -1 x =
òÅ‚ -1 + 4t
l1 : y = -1 + t , t " R l2 : y = -1 - 3t , t " R
ół ół
z = 2 + 3t z = 2 + t
"
73
Ä„ Ä„
; ,
6 3
Ä„ Ä„
A(2, 0, 1) Õ = ; B(1, 1 - 2) Õ =
3 6
(x - 1)2 + (y + 1)2 = 8 (x - 1)2 + (y - 1)2 = 4 (x + 1)2 + (y + 1)2 = 1 (x + 5)2 + (y + 5)2 = 25
(x - 1)2 + (y - 1)2 = 18
x2 (y-2)2
+ = 1
4 2
y2
x2
+ = 1
16 9
(x-1)2 (y+2)2
+ = 1, (P (1, -2),
4 9
(x-4)2 (y-2)2
+ = 1, µ = 0, 3
25 16
(x-2)2 (y-1)2
5
- = 1, µ =
16 9 4
"
(x-3)2 (y+3)2
10
- = 1, W1(-6, -3) W2(12, -3), µ =
81 9 3
2
"
2 2
x = 0, F (2, 2) y = 3, F (3, 8)