proving there exists a non zero polynomial for which B has a root

Question

For any square matrix B with entries in K, prove that there is a nonzero polynomial p ∈ K[t] which has B as a root.

For this question, we know it has something to do with linear dependency but are not sure how to start.

Answer 1

The $n\times n$ matrices for a vector space whose dimension is $n^2$. Consider the matrices $\operatorname{Id},B,B^2,\ldots,B^{n^2}$. You have here $n^2+1$ polynomials. So, they are linearly dependent, which means that there are scalars $\alpha_0,\alpha_1,\ldots\alpha_{n^2}\in K$, not all equal to $0$, such that$$\alpha_0\operatorname{Id}+\alpha_1B+\alpha_2B^2+\cdots+\alpha_{n^2}B^{n^2}.$$