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Say I have a system of equations (say 3),with a parameter k.Would excluding the zero values of the determinant of A be enough in order to find the values of k for which the system has a unique solution?

I know its true if its Ax = 0,but im not sure about the case where its Ax = b (would that work for any b?).

Is the only way to row reduce it? Because row reducing with parameters is kind of hard.
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What do you mean by excluding the values where the determinant is 0

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