For any field $k$, let $M_n(k)$ denote the ring of $n\times n$ matrices over $k$. Out of curiosity, do we know a way to relate polynomials (especially irreducible polynomials) in $k[x]$ with polynomials in $M_n(k)[x]$?
What about roots of polynomials in $M_n(k)[x]$? If a diagonalizable matrix $A$ is a root of a polynomial $f$ in $M_n(k)[x]$, then it's eigenvalues are roots of polynomials in $k[x]$ derived from $f$. Can we say more?
Your first question is not reasonable.
It does not make sense to talk about irreducible polynomials over $M_n(k)$, the reason being that $M_n(k)$ is not factorial and so is $M_n(k)[x]$. Even worst, it does not make sense to talk about divisibility in $M_n(k)[x]$, since $M_n(k)$ is not an integral domain, which implies that $M_n(k)[x]$ is not an integral domain too.