nontrivial solution
when Ax = 0 has a nonzero x solution that satisfies Ax = 0
In mathematics, a nontrivial solution is a solution to a problem that is not self-evident or obvious. It is a solution that cannot be obtained simply by inspection or common sense reasoning.
For example, consider the equation x^2 = 0. The trivial solution to this equation is x = 0. However, there is also a nontrivial solution, namely x = -0. This solution is not immediately apparent and requires careful analysis of the equation.
Another example is the problem of finding the roots of a quadratic equation ax^2 + bx + c = 0. The trivial solution of this problem is when a, b, and c are such that the discriminant b^2 – 4ac is zero, in which case there is one real root. However, the nontrivial solutions occur when the discriminant is negative, in which case there are two complex roots.
In general, a nontrivial solution is one that takes some effort to find and is not immediately obvious. These solutions can often involve complex mathematical reasoning and advanced techniques.
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