Matrix Polynomial with Non-Zero Coefficients

Question

Let $A ∈ M_{n \times n}(F)$ for some field $F$. Prove there is some non-zero polynomial $p(x)$ such that $p(A) = 0$.

I approached this question by dealing with each entry of the matrix and equating it to the respective $0$ on the $0$ matrix but I am stuck now as to what I should do next. Also the question does not allow the use of Cayley Hamilton Theorem.

Related: https://en.m.wikipedia.org/wiki/Cayley–Hamilton_theorem — Martin R, Dec 24 '20 at 00:18

score 1 · Accepted Answer · answered Dec 23 '20 at 23:55

1

$I,A,A^{2},...,A^{n^{2}+1}$ cannot all be linearly independent because the dimension of the space of $n \times n$ matrices is $n^{2}$. Hence, some non-trivial linear combnation of these must vanish.

answered Dec 23 '20 at 23:55

Kavi Rama Murthy

311,013

Thanks ! Makes lot of sense – Rij Dec 23 '20 at 23:58

Matrix Polynomial with Non-Zero Coefficients

1 Answers1