This is the last followup question to my other question. Here I have a matrix $M$ whose kernel is the field $\mathbb{F_p}$. I'm not sure what this expression is saying or why it is important. My understanding is that $\mathbb{F_p}$ is a field that contains individual non-vector elements so how can this possibly be in the kernel of matrix $M$?
For example, let us say that I have this matrix in $\mathbb{F_3}$ when is that matrix's kernel $\mathbb{F_3}$ ?