Exercise Without any computation, say as much as you can about the existence and the uniqueness for the solutions of the following systems?
1)
2x_{1} | + 2x_{3} | + x_{4} | = | 2 | |
x_{1} | + x_{2} | - x_{3} | = | -1 | |
x_{2} | - 2x_{3} | + x_{4} | = | -2 |
Answer The system is consistent because of the obvious solution x_{1} = x_{2} = x_{4} = 0, x_{3} = 1. The solution is not unique (so there are infinitely many) because 3 (number of rows/equations) < 4 (number of columns/variables).
2)
2x_{1} | + 2x_{3} | = | 2 | |
x_{1} | + x_{2} | - x_{3} | = | -1 |
x_{2} | - 2x_{3} | = | -2 |
Answer The system is consistent for the the obvious solution x_{1} = x_{2} = 0, x_{3} = 1. However, since the number of rows/equations (=3) is equal to the number of columns/variables (=3), we cannot say much about uniqueness of the solution without computation.
3)
x | + y | = | 4 |
2x | + 2y | = | 3 |
3x | + 3y | = | 2 |
4x | + 4y | = | 1 |
Answer The system is not consistent.
4)
x_{1} | - x_{2} | + x_{3} | + 2x_{5} | = | a | |
- 2x_{2} | + 5x_{3} | + 2x_{4} | + 2x_{5} | = | b | |
- 3x_{4} | + 7x_{5} | = | c | |||
- x_{5} | = | d |
Answer Note that the coefficient matrix is already in row echelon form, from which we see that the solutions always exist for all a, b, c, d. Moreover, the solution cannot be unique because 4 < 5.
5)
1.23x_{1} | - 3.75x_{2} | + 0.02x_{3} | = | a |
- 4.57x_{2} | + 3.34x_{3} | = | b | |
- 8.32x_{1} | - 3.05x_{2} | - 1.01x_{3} | = | c |
+ 7.98x_{3} | = | d | ||
4.71x_{1} | + 5.64x_{2} | + 11.26x_{3} | = | e |
Answer Since 5 > 3, the system does not have solutions for any right side (i.e., we can find some specific values of a, b, c, d, e so that the system has no solution). Moreover, if the system has solutions (for some choices of right side, such as a = b = c = d = e = 0), then the solutions are also the solutions of the following subsystem
1.23x_{1} | - 3.75x_{2} | + 0.02x_{3} | = | a |
- 4.57x_{2} | + 3.34x_{3} | = | b | |
+ 7.98x_{3} | = | d |
Since the solution of the subsystem is clearly unique, we conclude the solution for the original system is also unique.