# The order of the variable in the inverse matrix

Given a matrix $$C=A_0+A_1x$$ with the variable $$x$$ and two matrix $$A_0, A_1$$, the elements in the inverse matrix $$C^{-1}$$ can be represented by the quotient of two polynomials of x with highest orders $$k_1$$ and $$k_2$$. Can I determine the relation $$k_1\geq k_2$$?

No. Let $$C^{-1}=[c'_{i,j}]$$. If the matrices have dimension $$n$$, then $$c'_{i,j}=u_{i,j}/\det(C)$$ where $$u_{i,j}=\pm$$ the cofactor of $$(i,j)$$ has (in general) degree $$n-1$$; on the other hand, $$\det(C)$$ has, in general, degree $$n$$.
Your business is to consider (at least) an example with $$n=2$$. A little hard work never hurt anyone.