Советы по поводу особых ситуаций.
Если после зануления первого столбца получается a22 = 0, то на второе место переставляем другую строку, в которой второй элемент не равен нулю:
0 a 0 0 .
0 b 0 b
Если во втором столбце все числа «нехорошие» можно сделать дополнительное действие, с тем, чтобы получить среди них единицу. Например:
0 6
0 1
0 4 0 4 .
0 5 0 5
Задания для самостоятельного решения.
Найти общее решение однородной системы линейных уравнений и выписать фундаментальную систему решений. Сделать проверку.
1.
x1
+ x2
+ 2x3
– 5x4
– x5
= 0
2x1 + 3x2 + 2x3 –10x4 = 0
–3x1 – 4x2 – 3x3 +14x4 + x5 = 0
2x1 + 4x2 – 2x3 – 8x4 + 2x5 = 0
2 . x1+ 2x2 + 2x3 – 4x4 – 5x5 = 0
2x1+ 5x2 + 5x3 – 9x4–13x5 = 0
–3x1– 4x2 – 3x3 + 9x4 + 7x5 = 0
–2x1 – x2 + 4x4 – x5 = 0
3 . x1+ 2x2 – 2x3 + x4 + x5 = 0
2x1+ 5x2 – 2x3 + x4 – x5 = 0
3x1+ 4x2 – 9x3 + 5x4 + 8x5 = 0
–2x1– 2x2 + 9x3 – 4x4 – 9x5 = 0
4 . –2x1 – x2 + 4x3 – x4 – 3x5 = 0
x1 – 2x2 + 9x4 + 2x5 = 0
2x1 – 2x3 + 2x4 + 2x5 = 0 (:2)
3x1 + x2 – 2x3 – 4x4 + x5 = 0
5 . x1 + 3x2 – 2x3 + 5x5 = 0
3x1+ 4x2 – x3 + 5x4 = 0
–2x1 – x2 + 2x3 – 8x4 – 4x5 = 0
x1 + 4x2 – 3x3 – x4 + 8x5 = 0
6 . 2x1 – x2 + x3 – x4 – 4x5 = 0
– 4x1 + 3x2 + x3 – 5x4 + 8x5 = 0
4x1 – x2 – 3x3 – x4 = 0
6x1 – 2x2 – 2x3 – 2x4 – 4x5 = 0
7 . x1+ 2x2 – x3 – 2x4 + 2x5 = 0
5x1+ 3x2 – 3x3 – 6x4 – 4x5 = 0
–2x1+ 3x2 + x3 – 5x4 +10x5 = 0
x1+ 4x2 – 3x3 + 4x4 + 6x5 = 0
8 . x1 + x2 + x3 – 2x4 + 2x5 = 0
–2x1 + x3 – 3x4 – x5 = 0
3x1+ 4x2 + 5x3 – 10x4 +7x5 = 0
–2x1+ x2 + 3x3 – 7x4 = 0
9 . 2x1 + 2x3 + 2x4 + 2x5 = 0
2x1+ x2 + 2x3 + 4x4 + 2x5 = 0
3x1+ 2x2 + 7x4 + 6x5 = 0
– 4x1–3x2 –3x3 –10x4 – 5x5 = 0
1 0. 2x1 + 3x2 +2x3 + 3x4 + 3x5 = 0
2x1+ 2x2 + 2x4 + 2x5 = 0
3x1+ x2 – 2x3 + 3x4 – x5 = 0
–5x1 – 2x2 + 4x3 – 4x4 = 0
11. x1 + x2 + 2x3 – x4 – 5x5 = 0
x1 + 3x2 – 4x3 + 3x4 – 3x5 = 0
3x1 + 4x2 + 3x3 – x4 – 14x5 = 0
2x1 + 3x2 + 2x3 – 10x5 = 0
1 2. x1+ 2x2 + 2x3 – 5x4 – 4x5 = 0
3x1 + 3x2 + 2x3 – 4x4 – 8x5 = 0
–2x1– 5x2 – 5x3 + 13x4 + 9x5 = 0
3x1+ 4x2 + 3x3 – 7x4 – 9x5 = 0
1 3. x1+ 2x2 – x3 + x5 = 0
–2x1– 2x2 + 9x3 – 9x4 – 4x5 = 0..
2x1+ 5x2 – 2x3 – x4 + x5 = 0..
3x1+ 4x2 – 9x3 + 8x4 + 5x5 = 0..
1 4. x1– 2x2 + 2x4 + 9x5 = 0
2x1 + x2 – 4x3 + 3x4 + x5 = 0
3x1 + x2 – 2x3 + x4 – 4x5 = 0
2x1 – 2x3 + 2x4 + 2x5 = 0 (:2)
1 5. x1 + 3x2 – 2x3 + 5x4 = 0
x1 + 4x2 –3x3 + 8x4 – x5 = 0
2x1 + x2 – 2x3 + 4x4 + 8x5 = 0
3x1 + 4x2 – x3 + 5x5 = 0
1 6 . 2x1 – x2 + x3 + 4x4 – x5 = 0
4x1 – 3x2 – x3 + 8x4 + 5x5 = 0
6x1 – 2x2 – 2x3 + 4x4 – 2x5 = 0
– 4x1 + x2 + 3x3 + x5 = 0
1 7. x1+ 2x2 – x3 + 2x4 – 2x5 = 0
2x1– 3x2 – x3 –10x4 + 5x5 = 0
x1+ 4x2 – 3x3 + 6x4 + 4x5 = 0
5x1+ 3x2 – 3x3 – 4x4 – 6x5 = 0
1 8. – x1 + x2 + 2x3 + x4 – 5x5 = 0
–2x1+ x2 + 3x3 – 7x5 = 0
2x1 – x3 + x4 + 3x5 = 0
3x1+ 4x2 + 5x3 + 7x4 –10x5 = 0
1 9. 2x1+ x2 + 2x3 + 2x4 + 4x5 = 0
3x1+ 2x2 + 6x4 + 7x5 = 0
4x1+3x2 +3x3 +5x4 + 10x5 = 0
2x1 + 2x3 + 2x4 + 2x5 = 0
2 0. x1 + 2x2 + 2x3 – 2x4 + 2x5 = 0
2x1 + 3x2 +2x3 – 3x4 + 3x5 = 0
2x1+ 2x2 – 2x4 + 2x5 = 0
5x1+ 2x2 – 4x3 + 4x5 = 0
2 1. x1+ 2x2 – 5x3 + 2x5 = 0
4x1+ 9x2 –17x3 +5x4+13x5 = 0
– x1– 3x2 + 2x3 – 3x4 – 5x5 = 0
–2x1– 5x2 + 7x3 – 3x4 – 7x5 = 0
2 2. –3x1–7x2 + 4x3 – 9x4 + 5x5 = 0
x1 + 3x2 + 3x4 + x5 = 0
–5x1–12x2 + 8x3 –19x4 + x5 = 0
– 2x2 – 7x3 + 6x4 + x5 = 0
2 3. x1–3x2 + x3 + 5x4 + 4x5 = 0
– x1 + 3x2 + x3 – 7x4 = 0
2x1– 6x2 + 5x3 + 9x4 +10x5 = 0
3x1– 9x2 – x3 +14x4 +14x5 = 0
2 4. x1 + 2x3 + x4 – x5 = 0
3x1 – x2 + 9x3 + 4x4 + 2x5 = 0
2x1 + 2x2 – 2x3 + 3x4 –15x5 = 0
– 2x1– 3x2 + 5x3 – 5x4 +23x5 = 0
2 5. x1 + 2x2 – 2x3 + x5 = 0
3x1 + 7x2 – 4x3 + x4 + 4x5 = 0
– x1 + 5x3 + 2x4 – 4x5 = 0
– 2x1 – x2 +11x3 + 3x4 + 6x5 = 0
2 6. x1+ 3x2 – 2x3 + 5x4 + 3x5 = 0
2x1+ 2x2 – 8x3 – 2x4 + 2x5 = 0
x1+ 2x2 – 5x3 + 2x4 = 0
4x1+ 9x2 –17x3 +11x4+3x5 = 0
2 7. x1 + 3x2 + x4 + 3x5 = 0
3x1+7x2 – 4x3 – 5x4 + 9x5 = 0
x1 + x2 – 7x3 + 2x4 + 9x5 = 0
5x1+12x2 – 8x3 – x4 +19x5 = 0