 Minimal polynomial of an invertible matrix

A square matrix $A$ is invertible exactly when its minimal polynomial has nonzero constant term. If $m_A(t)=tq(t)$, then $Aq(A)=0$, and invertibility would contradict the minimality of $m_A$. Conversely, if $m_A(t)=tq(t)+c$ with $c\ne 0$, then $A[-q(A)/c]=I$.