îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
0 1 3 -2 1 -5 1 1
1 -1 0
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A = -1 2 1 , B = 0 -3 -3 , C = -2 0 , D = .
-1 0 2
1 0 -1 -2 1 1 1 1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 1 1 -1 1 1 2 -2
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
C + 2 · DT = -2 0 + 2 · -1 0 = -2 0 + -2 0 =
1 1 0 2 1 1 0 4
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 + 2 1 + (-2) 3 -1
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
= -2 + (-2) 0 + 0 = -4 0 ,
1 + 0 1 + 4 1 5
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
-2 1 -5 0 1 3 -2 - 0 1 - 1 -5 - 3
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
B - A = 0 -3 -3 - -1 2 1 = 0 - (-1) -3 - 2 -3 - 1 =
-2 1 1 1 0 -1 -2 - 1 1 - 0 1 - (-1)
îÅ‚ Å‚Å‚
-2 0 -8
ðÅ‚ ûÅ‚
= 1 -5 -4 ,
-3 1 2
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
-2 1 -5 0 1 3
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
B · A = 0 -3 -3 · -1 2 1 =
-2 1 1 1 0 -1
îÅ‚ Å‚Å‚
-2·0+1 ·(-1)+(-5)·1 -2·1+1·2+(-5)·0 -2·3+1·1+(-5) · (-1)
ðÅ‚
= 0·0+(-3) ·(-1)+(-3)·1 0·1+(-3) ·2+(-3)·0 0·3+(-3) ·1+(-3)·(-1)ûÅ‚=
(-2)·0+1 ·(-1)+1·1 (-2)·1+1 ·2+1·0 (-2)·3+1 ·1+1·(-1)
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
-6 0 0 1 0 0
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
= 0 -6 0 = -6 · 0 1 0 = -6 · I.
0 0 -6 0 0 1
2 -3
det = 2 · 4 - 5 · (-3) = 8 + 15 = 23,
5 4
îÅ‚ Å‚Å‚
-2 3 1
ðÅ‚ ûÅ‚
det 1 4 2 = -2 · 4 · 3+1 · 2 · 1+(-1) · 3 · 2-(-1) · 4 · 1-2 · 2 · (-2)-3 · 1 · 3 =
-1 2 3
= -24 + 2 - 6 + 4 + 8 - 9 = -25
îÅ‚ Å‚Å‚
-2 3 -8
ðÅ‚ ûÅ‚
A = 1 -5 -4
-3 1 0
-2 3
A23 = (-1)2+3 · M23 = -M23 = - = -(-2 + 9) = -7.
-3 1
îÅ‚ Å‚Å‚
2 3 1 1
ïÅ‚ śł
4 1 1 2
ïÅ‚ śł
A = .
ðÅ‚ ûÅ‚
1 1 0 2
3 2 3 1
2 3 1 1
4 1 1 2
det A = = a11A11 + a21A21 + a31A31 + a41A41 =
1 1 0 2
3 2 3 1
1 1 2 3 1 1 3 1 1
= 2 · (-1)1+1 · 1 0 2 + 4 · (-1)2+1 · 1 0 2 + 1 · (-1)3+1 · 1 1 2 +
2 3 1 2 3 1 2 3 1
3 1 1
+ 3 · (-1)4+1 · 1 1 2 = 2(6 + 4 - 6 - 1) - 4(3 + 4 - 18 - 1) +
1 0 2
+ (3 + 3 + 4 - 2 - 18 - 1) - 3(6 + 2 - 1 - 2) = 6 + 48 - 11 - 15 = 28
îÅ‚ Å‚Å‚
2 3 1 1
ïÅ‚ śł
4 1 1 2
ïÅ‚ śł
A = .
ðÅ‚ ûÅ‚
1 1 0 2
3 2 3 1
 + 2 · 
 - 
2 3 1 1 1 3 1 1 1 3 1 1
·
·
4 1 1 2 3 1 1 2 0 -8 -2 -1
det A = = = =
1 1 0 2 1 1 0 2 0 -2 -1 1
3 2 3 1 0 2 3 1 0 2 3 1
1 -3 1 1 1 -3 1 1
·
·
0 -2 -2 -1 0 -2 -2 -1
= = =
0 -8 -1 1 0 0 7 5
0 -4 3 1 0 0 7 3
1 -3 1 1
-2 -2 -1
0 -2 -2 -1
= = 1 · (-1)1+1 0 7 5 =
0 0 7 5
0 0 -2
0 0 0 -2
7 5
= 1 · (-2)(-1)1+1 = 1 · (-2) · 7(-1)1+1 -2 =
0 -2
= 1 · (-2) · 7 · (-2) = 28.
îÅ‚ Å‚Å‚
1 0 1
ðÅ‚ ûÅ‚
A = -1 -1 0 .
0 1 2
A
1 0 1
det A = -1 -1 0 = -2 - 1 = -3 = 0.

0 1 2
A-1
-1 0
A11 = (-1)1+1 · = -2,
1 2
-1 0
A12 = (-1)1+2 · = 2,
0 2
-1 -1
A13 = (-1)1+3 · = -1,
0 1
0 1
A21 = (-1)2+1 · = 1,
1 2
1 1
A22 = (-1)2+2 · = 2,
0 2
1 0
A23 = (-1)2+3 · = -1,
0 1
0 1
A31 = (-1)3+1 · = 1,
-1 0
1 1
A32 = (-1)3+2 · = -1,
-1 0
1 0
A33 = (-1)3+3 · = -1.
-1 -1
îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚T îÅ‚ Å‚Å‚
2
-1 -1
-2 2 -1 -2 1 1
3 3 3
1 1 1
ïÅ‚
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A-1 = [Aij]T = 1 2 -1 = 2 2 -1 = -2 -2 1 śł .
ðÅ‚ ûÅ‚
3 3 3
det A -3 -3
1 1 1
1 -1 -1 -1 -1 -1
3 3 3
AA-1 A-1A
îÅ‚ Å‚Å‚
1 0 1
ðÅ‚ ûÅ‚
A = -1 -1 0 .
0 1 2
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 0 1 1 0 0 1 0 1 1 0 0
w2 + w1 w2 · (-1)
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A I = -1 -1 0 0 1 0 - 0 -1 1 1 1 0 -
0 1 2 0 0 1 0 1 2 0 0 1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 0 1 1 0 0 1 0 1 1 0 0
1
w3 - w2 w3 ·
3
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
0 1 -1 -1 -1 0 - 0 1 -1 -1 -1 0 -
0 1 2 0 0 1 0 0 3 1 1 1
îÅ‚ Å‚Å‚
îÅ‚ Å‚Å‚
2
w2 + w3
1 0 0 -1 -1
1 0 1 1 0 0
3 3 3
w1 - w3
ïÅ‚
ðÅ‚ ûÅ‚
0 1 -1 -1 -1 0 - 0 1 0 -2 -2 1 śł = I A-1 .
ðÅ‚ ûÅ‚
3 3 3
1 1 1
1 1 1
0 0 1
0 0 1
3 3 3
3 3 3
AX = B
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 0 1 3 3
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
A = -1 -1 0 B = 6 0 .
0 1 2 -9 -3
A X
A-1
A-1AX = A-1B
IX = A-1B
X = A-1B
A-1
îÅ‚ Å‚Å‚
2
-1 -1 îÅ‚ 3 3 Å‚Å‚ îÅ‚ 3 3 Å‚Å‚
3 3 3
ïÅ‚
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
X = -2 -2 1 śł · 6 0 = -9 -3
ðÅ‚ ûÅ‚
3 3 3
1 1 1
-9 -3 0 0
3 3 3
îÅ‚ Å‚Å‚
1 0 1
ðÅ‚ ûÅ‚
A = -1 -1 0 .
0 1 2
1 0 1
det A = -1 -1 0 = -2 - 1 = -3 = 0,

0 1 2
A = 3
îÅ‚ Å‚Å‚
1 1 2
ïÅ‚ śł
2 -1 1
ïÅ‚ śł
B = .
ðÅ‚ ûÅ‚
3 0 3
-1 -1 -2
B 4 × 3
 B
B = 3
1 1 2
2 -1 1 = -3 + 3 + 6 - 6 = 0.
3 0 3
1 1 2
2 -1 1 = 2 - 2 - 1 - 2 - 1 + 4 = 0.
-1 -1 -2
1 1 2
3 0 3 = -6 - 3 + 3 + 6 = 0.
-1 -1 -2
2 -1 1
3 0 3 = -3 + 3 + 6 - 6 = 0.
-1 -1 -2
B
B < 3
1 1
= -3 = 0 B = 2

2 -1
îÅ‚ Å‚Å‚
1 1 2
ïÅ‚ śł
2 -1 1
ïÅ‚ śł
B = .
ðÅ‚ ûÅ‚
3 0 3
-1 -1 -2
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 1 2 1 1 2
w4 + w1
ïÅ‚ śł ïÅ‚ śł
w3
2 -1 1 - (w1 + w2) w3, w4
2 -1 1
ïÅ‚ śł ïÅ‚ śł
B = = =
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
3 0 3 0 0 0
-1 -1 -2 0 0 0
k3
1 1 2 - (k1 + k2) k3
1 1 0 1 1
= = = = 2.
2 -1 1 2 -1 0 2 -1
Å„Å‚
x + y + z = 0
òÅ‚
2x - y - z = -3 .
ół
4x - 5y - 3z = -7
îÅ‚ Å‚Å‚
1 1 1
ðÅ‚ ûÅ‚
A = 2 -1 -1
4 -5 -3
1 1 1
W = det A = 2 -1 -1 = 3 - 10 - 4 + 4 - 5 + 6 = -6 = 0.

4 -5 -3
det A = 0

0 1 1
Wx = -3 -1 -1 = 15 + 7 - 7 - 9 = 6
-7 -5 -3
1 0 1
Wy = 2 -3 -1 = 9 - 14 + 12 - 7 = 0
4 -7 -3
1 1 0
Wz = 2 -1 -3 = 7 - 12 - 15 + 14 = -6
4 -5 -7
Å„Å‚
6
ôÅ‚
ôÅ‚
x = = -1
ôÅ‚
ôÅ‚
ôÅ‚ -6
ôÅ‚
òÅ‚
0
y = = 0
ôÅ‚ -6
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚ -6
ôÅ‚
ół
z = = 1
-6
3x - y + z = 2
.
6x - 2y + 2z = 1
3 -1 1 3 -1 1 2
A = , U = .
6 -2 2 6 -2 2 1
2×3 2×4
A U
w2
3 -1 1 - 2 · w1 w2
3 -1 1
A = = = 3 -1 1 = 1,
6 -2 2 0 0 0
k1 + 3 · k2
k3 + ·k2 k1, k3 -1 2
3 -1 1 2 0 -1 0 2
U = = = = 2.
6 -2 2 1 0 -2 0 1 -2 1
A = U

Å„Å‚
x
òÅ‚ - y + z = 0
2x - 2y + z = 1 .
ół
3x - 3y + 2z = 1
îÅ‚ Å‚Å‚
1 -1 1
ðÅ‚ ûÅ‚
A = 2 -2 1 ,
3 -3 2
1 -1 1 1 -1 1
w3 - w1 - w2
det A = 2 -2 1 = 2 -2 1 = 0.
3 -3 2 0 0 0
A U A
-1 1
= -1 + 2 = 1 = 0 A = 2

-2 1
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
1 -1 1 0 1 -1 1 0
w3 - w1 - w2
1 -1 1 0
ðÅ‚ ûÅ‚ ðÅ‚ ûÅ‚
U = 2 -2 1 1 = 2 -2 1 1 = =
2 -2 1 1
3 -3 2 1 0 0 0 0
-1 1
= = 2
-2 1
r = A = U = 2 n = 3
-1 1
n - r = 3 - 2 = 1
-2 1
-y + z = -x
.
-2y + z = 1 - 2x
x t t " R
-y + z = -t
.
-2y + z = 1 - 2t
-1 1
W = = 1,
-2 1
-t 1
Wy = = -t + 2t - 1 = t - 1,
1 - 2t 1
-1 -t
Wz = = 2t - 1 - 2t = -1.
-2 1 - 2t
Å„Å‚
ôÅ‚ x = t
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
òÅ‚
t - 1
y = = t - 1
t " R
1
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚ -1
ôÅ‚
ół
z = = -1
1


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