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Like I said, "we have to assume b cannot be zero" for there to be any "bad" value of a. It could just be an issue of ambiguous wording (as written, it sounds like it is stating a conclusion about the system for arbitrary b, whereas maybe it is meant to be taken as an extra property that allows us to deduce that b is nonzero).
But even with b != 0, the only possible bad value of a is -1 (and this is bad for all nonzero b). I'm not sure how you arrived at a = 2 but in that case there's actually infinitely many solutions for any given value of b (IOW a = 2 is arguably the most wrong answer possible here); for example (x, y, z) = (1, -4, 1) when b = -6. More specifically, if a = 2 and given any value of b, we are free to pick any value of x, and then take z = -x-b/3, and y = 2b/3.
So the condition stated by the problem (that there's more than one value of a which leads to a system with no solutions) is actually an impossible condition. It's a bad statement because it probably leads people to believe wrong things about the relationship between determininant and whether system has no solutions.
Linear Algebra Foundations #8 - Systems of Equations
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Like I said, "we have to assume b cannot be zero" for there to be any "bad" value of a. It could just be an issue of ambiguous wording (as written, it sounds like it is stating a conclusion about the system for arbitrary b, whereas maybe it is meant to be taken as an extra property that allows us to deduce that b is nonzero).
But even with b != 0, the only possible bad value of a is -1 (and this is bad for all nonzero b). I'm not sure how you arrived at a = 2 but in that case there's actually infinitely many solutions for any given value of b (IOW a = 2 is arguably the most wrong answer possible here); for example (x, y, z) = (1, -4, 1) when b = -6. More specifically, if a = 2 and given any value of b, we are free to pick any value of x, and then take z = -x-b/3, and y = 2b/3.
So the condition stated by the problem (that there's more than one value of a which leads to a system with no solutions) is actually an impossible condition. It's a bad statement because it probably leads people to believe wrong things about the relationship between determininant and whether system has no solutions.