Existence of some special primitive normal elements over finite fields

In this article, we establish a sufficient condition for the existence of a primitive element α∈Fq such that for any matrix (abc0de)∈M2×3(Fq) of rank 2, the element (aα2+bα+c)/(dα+e) is a primitive element of Fq, where q=2k for some positive integer k. We also give a sufficient condition for the existence of a primitive normal element α∈Fqn over Fq such that (aα2+bα+c)/(dα+e) is a primitive element of Fqn for every matrix (abc0de)∈M2×3(Fqn) of rank 2.