Finding parameter $x$ of a system of linear equation $\boldsymbol{A}(x)\,\boldsymbol{c}=\boldsymbol{0}$ leading to non-trivial solutions

Question

I am wondering, which possiblities exist to find the parameter $x$ in a system of linear equations $$\boldsymbol{A}(x)\,\boldsymbol{c}=\boldsymbol{0}$$ leading to a non-trivial solution $\boldsymbol{c}$. The dependency of $\boldsymbol{A}$ of $x$ should be arbitrary (non-linear).

I know that I can use $\mathrm{det}(\boldsymbol{A}(x))=0$ to find the values of $x$ leading to a non-trivial solution, but that seems to be rather instable from a numerical point, especially if the matrix dimension becomes larger. Are there other conditions, which can be used in numerical algorithms?

Edit: $\boldsymbol{A}(x)$ can be complex valued and is square.

Edit 2: If it narrows the problem a bit, the entries of $\boldsymbol{A}(x)$ are of the form $a_1\,x^n\sin(a_2x)$, $b_1\,x^n\cos(b_2x)$, $c_1\,x^n\exp(c_2x)$ with $a_1$, $a_2$, $b_1$, $b_2$, $c_1$, $c_2$ arbitrary complex constants.

Edit 3: $x$ is generally also complex valued