This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The main\ud result in the first part of this paper is a theorem which gives sufficient conditions for the ideal of minors of a matrix to be prime. This theorem is general enough to include interesting examples, such as the ideal of maximal\ud minors of catalecticant matrices and their generalisations discussed in the second part of the paper.\ud The second aim of this paper is to settle a specific problem raised by David Eisenbud and Frank-Olaf Schreyer on the primary decomposition of an ideal of maximal minors. We solve this problem by applying the theorem above together with some ad-hoc techniques.\u
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