The order of the variable in the inverse matrix

Question

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 $k_1\geq k_2$?

Answer 1

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.