- #1
4Fun
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Homework Statement
Prove that the linear system of the form:
[tex]
a_{1,1}x + a_{1,2}y = b_{1}\\
a_{2,1}x + a_{2,2}y = b_{2}
[/tex]
has exactly one solution in x and y if [itex]a_{1,1}a_{2,2} \neq a_{1,2}a_{2,1}[/itex]
The Attempt at a Solution
I proved the contrapositive:
Assume that the given system does not have exactly one solution in x and y, then it either has no solution or it has infinitely many solutions.
Case 1: Assume that there is no solution to this system. Since it is a system of two equations, I can refer to a geometric explanation. Two equations do not intersect if they are parallel, i.e. their slopes are equal and b1 != b2. For their slopes to be equal it has to be the case that [itex]a_{1,1}a_{2,2} = a_{1,2}a_{2,1}[/itex].
Case 2: Assume there are infinitely many solutions, in that case the equations are the same and their slopes are equal, therefore again fulfilling [itex]a_{1,1}a_{2,2} = a_{1,2}a_{2,1}[/itex].
Now I think the proof is correct, but I don't like to rely on geometric intuition, if it would have been a system of 3 equations I wouldn't be able to prove this type of problem. I tried applying the Gauss algorithm to this system, but it did not yield anything useful.
Could anybody give me a hint for another approach please?