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Задать вопрос экспертам

Консультация № 161308

24.02.2009, 18:17

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Образование

Здравствуйте многоуважаемые эксперты.Помогите пожулуйста.Мой вопрос: Дана система линейных уравнений. Доказать ее совместность и решить двумя способами: 1) методом Гаусса; 2) средствами матричного исчисления.
1*Х1-4Х2-2Х3=-3
2*3Х1+Х2+Х3=5
3*3Х1-5Х2-6Х3=-9

Обсуждение

Неизвестный

25.02.2009, 11:40

общий

это ответ

Здравствуйте, Kotr67!
|1...-4...-2|
|3....1....1|=-6+30-12+6+5-72=24-6-67=24-73=-49=d не равно нулю, значит система уравнений совместна
|3..-5....-6|
1) Метод Гаусса
(1...-4...-2|-3)..(1...-4...-2|-3)....(1...0...-2|-3)...(1..0....0|1)
(3....1.....1|5)~(0...13....7|14)~(0...0....7|14)~(0...0....1|2)
(3...-5..-6|-9)...(0....7....0|0).....(0....1....0|0)....(0...1....0|0)
x1=1
x2=0
x3=2

2) Метод Крамера
d=-49 (см. начало решения)
|-3...-4...-2|
|5.....1....1|=18+50+36-18-15-120=68+18-135=86-135=-49=d1
|-9..-5....-6|
|1...-3...-2|
|3.....5....1|=-30+54-9+30+9-54=0=d2
|3..-9....-6|
|1...-4...-3|
|3.....1....5|=-9+45-60+9+25-108=36-51-83=36-134=-98=d3
|3..-5....-9|
x1=d1/d=-49/(-49)=1
x2=d2/d=0/(-49)=0
x3=d3/d=-98/(-49)=2

3) AX=B
X=A^-1B
(1...-4...-2)
(3....1....1)
(3..-5....-6)
A11=(-1)¹⁺¹(-6+5)=-1
A12=(-1)¹⁺²(-18-3)=21
A13=(-1)¹⁺³(-15-3)=-18
A21=(-1)²⁺¹(24-10)=-14
A22=(-1)²⁺²(-6+6)=0
A23=(-1)²⁺³(-5+12)=-7
A31=(-1)³⁺¹(-4+2)=-2
A32=(-1)³⁺²(1+6)=-7
A33=(-1)³⁺³(1+12)=13
(-1....-14.....-2)
(21......0.....-7)*1/(-49)=A^-1
(-18....-7....13)
X=A^-1B=
=-1/49*(-1....-14.....-2)..(-3)............(3-70+18)...............(-49)..(1)
.............(21......0.....-7)*.(5)=-1/49*(-63+0+63)=-1/49*(0)...=(0)
.............(-18....-7....13)..(-9)............(54-35-117)............(-98)..(2)

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