Let $A \in Mat(n,F)$ be a matrix. Consider the set of polynomials $J_A \subset F[t]$ such that $q(t) \in J_A$ if and only if $q(A) = 0.$ Show that $J_A$ is an ideal. The monic polynomial generating JA is called the minimal polynomial of A.
Question: What does $q(A) = 0$ mean?
It means that when you apply the polynomial $q$ to the matrix $A$ you get the zero matrix. For example, if $q(t)=t^2$, $r(t)=t^2+1$ and $A$ is the 2x2 matrix with a 1 in the upper-right corner and 0 everywhere else, then $q(A)=0$ and $r(A)=I$. (Coefficients in the polynomial are interpreted as scalar multiplication by the matrix and the constant term is interpreted as a scalar multiple of the identity matrix.)