Consider the following system of Fredholm integral equations with constant kernel matrix $$ f(x)=K(x)\int_{0}^{1}f(s)ds $$ where $K(x)\in C([0,1];M_{2\times 2}(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion )).$ For the scalar case, it is known by the Fredholm alternative that the above equation has a unique solution equals to zero if and only if $K(x)$ is different from $1$ for any $x\in [0,1]$.

My question is: What happens in the system case?. How I can check the spectrum condtion?. Thank you in advance.