The degree of the minimal polynomial of a matrix of dimension n is at most n²

Let A \in \mathsf{Mat}_n(F) be a matrix over a field F. Prove that the dimension of the minimal polynomial m_A(x) of A is at most n^2.

If V = F^n, note that the set \mathsf{End}_F(V) is isomorphic to \mathsf{Mat}_n(F) once we select a pair of bases. The set of all n \times n matrices over F is naturally an F-vector space of dimension n^2. Consider now the powers of A: 1 = A^0, A^1, A^2, …, A^{n^2}. These matrices, as elements of \mathsf{End}_F(V), are necessarily linearly dependent, so that \sum r_i A^i = p(A) = 0 for some r_i \in F. The minimal polynomial m_A(x) divides this p(x), and so has degree at most n^2.

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