# Why can't we have only one complex eigenvalue?

Let A be matrix. If $$bi$$ ($$b$$ is real) is eigenvalue then $$-bi$$ is also eigenvalue. Why can it alone exist? $$bi$$ is eigenvalue and $$-bi$$ is not eigenvalue.

My attempt at proving this: [eigenvalues are roots of characteristic equation. Any equation, complex roots exist in pair. But how to prove this?]

$$Ax = bi x$$ and say $$Ax \neq -bix \\ \implies Ax \neq -bix = - Ax \\ 2Ax \neq0$$