Exercise For what choices of the parameters are the following systems consistent?
1)
x | + 2y | = | a |
3x | + 4y | = | b |
Answer The augmented matrix is
[ | 1 | 2 | a | ] |
3 | 4 | b |
By -3[row 1] + [row 2], we get
[ | 1 | 2 | a | ] |
0 | -2 | b - 3a |
For any a and b, this is a row echelon form with no rows of the form [0 0 ≠0]. Therefore the system is consistent for all choice of a and b.
2)
x | + y | = | a |
2x | + 2y | = | b |
3x | + 3y | = | c |
4x | + 4y | = | d |
Answer The augmented matrix is
[ | 1 | 1 | a | ] |
2 | 2 | b | ||
3 | 3 | c | ||
4 | 4 | d |
By row operations, we get
[ | 1 | 1 | a | ] |
0 | 0 | b - 2a | ||
0 | 0 | c - 3a | ||
0 | 0 | d - 4a |
The condition for no rows of the form [0 0 0] is b = 2a, c = 3a, d = 4a. This is exactly the condition for the system to be consistent.
x_{1} | + x_{2} | + 2x_{3} | + x_{4} | = | 1 |
x_{1} | + 2x_{3} | = | 0 | ||
2x_{1} | + 2x_{2} | + 3x_{3} | = | h | |
x_{2} | + x_{3} | + 3x_{4} | = | h |
Answer The augmented matrix is
[ | 1 | 1 | 2 | 1 | 1 | ] |
1 | 0 | 2 | 0 | 0 | ||
2 | 2 | 3 | 0 | h | ||
0 | 1 | 1 | 3 | h |
By -[row 1] + [row 2] and 2-[row 1] + 3[row 2], we get
[ | 1 | 1 | 2 | 1 | 1 | ] |
0 | -1 | 0 | -1 | -1 | ||
0 | 0 | -1 | -2 | h - 2 | ||
0 | 1 | 1 | 3 | h |
By [row 2] + [row 4], we get
[ | 1 | 1 | 2 | 1 | 1 | ] |
0 | -1 | 0 | 0 | -1 | ||
0 | 0 | -1 | -2 | h - 2 | ||
0 | 0 | 1 | 2 | h - 1 |
By [row 3] + [row 4], we get
[ | 1 | 1 | 2 | 1 | 1 | ] |
0 | -1 | 0 | 0 | -1 | ||
0 | 0 | -1 | -2 | h - 2 | ||
0 | 0 | 0 | 0 | 2h - 3 |
The condition for the system to be consistent is 2h - 3 = 0. In other words, h = 3/2.
u | + 2v | = | 1 | ||
- 2x | + 2y | - u | + 4v | = | h |
x | - y | + u | - v | = | 1 |
Answer The augmented matrix is
[ | 0 | 0 | 1 | 2 | 1 | ] |
-2 | 2 | -1 | 4 | h | ||
1 | -1 | 1 | -1 | 1 |
By row operations, we get
[ | 1 | -1 | 0 | -3 | 0 | ] |
0 | 0 | 1 | 2 | 1 | ||
0 | 0 | 0 | 0 | h + 1 |
The condition for the system to be consistent is h = -1.
x_{1} | + x_{2} | - x_{3} | = | -2 |
x_{1} | - ax_{3} | = | -1 | |
x_{1} | + ax_{2} | = | -1 |
Answer The augmented matrix is
[ | 1 | 1 | -1 | -2 | ] |
1 | 0 | -a | -1 | ||
1 | a | 0 | -1 |
By -[row 1] + [row 2] and -[row 1] + [row 3], we get
[ | 1 | 1 | -1 | -2 | ] |
0 | -1 | 1 - a | 1 | ||
0 | a - 1 | 1 | 1 |
Then by (a+1)[row 2] + [row 3], we have
[ | 1 | 1 | -1 | -2 | ] |
0 | -1 | 1 - a | 1 | ||
0 | 0 | 2a - a^{2} | a |
If 2a - a^{2} ≠ 0, i.e., a ≠ 0 or 2, then the system is clearly consistent.
If a = 0, then the last row consists of zeros. Thus the system is also consistent.
If a = 2, then the last row is [0 0 0 2]. Thus the system is inconsistent.
x_{1} | + x_{2} | + x_{3} | + x_{4} | = | 1 |
x_{1} | + ax_{4} | = | 1 | ||
x_{2} | + ax_{3} | = | 1 |
Answer The augmented matrix is
[ | 1 | 1 | 1 | 1 | 1 | ] |
1 | 0 | 0 | a | 1 | ||
0 | 1 | a | 0 | 1 |
By row operations, we get
[ | 1 | 1 | 1 | 1 | 1 | ] |
0 | 1 | 1 | 1 - a | 0 | ||
0 | 0 | a - 1 | a - 1 | 1 |
The condition for the system to be consistent is a ≠ 1.
v | + w | = | a | |
u | + ax | = | 1 | |
u | + v | = | 0 | |
av | - w | = | 1 |
Answer The augmented matrix is
[ | 0 | 1 | 1 | a | ] |
1 | 0 | a | 1 | ||
1 | 1 | 0 | 0 | ||
0 | a | -1 | 1 |
By row operations, we get
[ | 1 | 0 | a | 1 | ] |
0 | 1 | 1 | a | ||
0 | 0 | - a - 1 | - a - 1 | ||
0 | 0 | 0 | - a^{2} + a + 2 |
The condition for the system to be consistent is -a^{2} + a + 2 = 0. In other words, the system has solutions only for a = -1 or 2.