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TW. KRONECKERA-CAPELLIEGO

1. Wyznaczyć rząd macierzy

	1 2 3		1	-2 -1	1
A =	4 5 6	B =	3 -	1 4	3
	7 8 9		1	-9 -8	1
	1 -1	0	2 1		2
C =	3 1	1	3 2	D =	3
	-1 -3	-1	1 0		4

'2 1-12'	F =	3 1	-1 2
3 1 0 2		6 2	-2 4

IX| +    + 2xj - x₄ = 3

2X| + Xj -3xj +3x₄ = 1 4X| + 3xj + x₃ + x₄ = 7

Odpowiedzi.

1. R(A)=2, R(B)=3, R(C)=2, R(D)=2,

0 3-5	4		-1 l‘
1 1 -1	1	H =	2 -2
2 1 3	-2		3 -3
			2 1 0
'2 4 5'
		N =	0 1 2
1 2 3
			1 0 2

2. Rozwiązać układy rownan

( x-2y + 3z = 1 a) 12x - 4y + 6: = 2 (3x-6y + 9c = 3

3x + 2y-c = 0 c) 2x - y + 3: = 0 x + 3y - 4: = 0

2x-3y = 6 c) x + 2y = 4 x-5y = 5

{x + 2y + 3: = 4 2x+ y- z = 3 3x + 3y + 2: = l

f *+y = \

d) - 2x + 2y = 2 (3.r + 3y = 3

[2x-3y = 6 f) ^3x+ y = 9 [x + 4y = 3

^g){

2x + 2y -8: = 1 5x -5y- lOe = 3

I x+2y+3z=4 h) 2x+ 4y+6z= 3 | - 2x+ y- z = l

fx, + x₂+x₃ + x₄=5 [x, + x₂+2x₃-x₄=3

I3xj -5xj +4x₄ =8

^xl^+x2^_x3 ^{+ x}4⁼⁵ 2X| - Xj + 3xj - 2x₄ = 2

R(E)=2, R(F)=1,    R(G)=3, R(H)=1,

R(M)=2, R(N)=3

2. a) ukl. nieozn. R(A) = R(U) - 1

x = 1 + 2t - 3s, y = /, z = s, t,s' R

b)    ukl. nieozn. R(A) = R(U) = 2

x = ^(2 + 5/), y = ±(5-7/), z = t;teR

c)    ukl. nieozn R( A) = R(U) = 2 x = -|r, y = yi, z = t; t eR

d)    ukl. nieozn. R( A) = R(U) - 1 x = l, y=\-l; 1 eR

e)    ukl. sprzeczny, R( A) = 2, R(U) = 3

f) ukl. ozn. x = 3, y-0

g)    ukl. nieozn. R( A) = R(U) = 2

x = t,y = ^⁽2a-¹⁴⁾,

z = ^(20t-ll);t£R

h)    ukl. sprzeczny, R( A) = 2, R(U) = 3

i)    R(A) = R(U) = 2

j)    R(A) = R(U) = 2

k)    R(A) = R(U) = 2

i)