Consider a unimodular square polynomial matrix $A(x)$ with elements defined over the polynomial ring $F[x]$ with coefficients from a finite Galois field. By the unimodular property, we know that the matrix has determinant equal to non-zero element, which also means that the inverse matrix $A(x)^{-1}$ exists and its elements are defined over $F[x]$ as well.
Could someone point me at a suitable method / approach for finding the inverse matrix $A(x)^{-1}$? I know that one could easily find the inverse over the field of rational functions $F(x)$ (for example by using Gauss-Jordan elimination, or LU factorization with full pivoting) but I need to find the inverse matrix with elements defined over the polynomial ring $F[x]$.
Thank you in advance!
Gauss-Jordan elimination of $A(x)$ yields a matrix in $F(x)$ that is inverse to $A(x)$. Because $A(x)$ is unimodular it has an inverse in $F[x]\subset F(x)$, so by uniqueness of the inverse, Gauss-Jordan elimination yields a matrix in $F[x]$.