Let $k$ be an algebraically closed field. Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of $A$.

Is $k[x_1, \ldots, x_p]/I$ a Cohen-Macaulay ring?

If no, does the additional assumption that $I$ is a radical (or prime or any other nice property) ideal help?

If yes, is there a good reference for that?