Vector/Value/As eigenvector/Linear system/Exercise

Let ${}K$ be a field, ${}c\in K$ and let ${}a={\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\in K^{n}$ denote a vector different from ${}0$. Establish an inhomogeneous system of linear equations such that its solution set is the set of all ${}n\times n$-matrices, for that ${}a$ is an eigenvector with eigenvalue ${}c$. What is special about this system, and what is the dimension of its solution set?

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