Does a matrix with polynomial coefficients have an inverse if its determinant is nonzero?

Question

If the determinant of a polynomial matrix $(a_{ij}(x))$ is nonzero, does it necessarily have an inverse?

I think it is not true because I think polynomials do not form a field, so a single polynomial may not have an inverse, so I think the matrix may not have inverse, too. But it is only a single thought. Could you give a strict statement and maybe a proof or a counterexample?

Answer 1

The determinant of the $1\times1$-matrix $[x]$ is nonzero, but it does not have an inverse.

In general a matrix with entries from a commutative ring $R$ is invertible if and only of its determinant is invertible in $R$. See this question and its answer for details.