I need to prove the following:
Let $A$ be a square matrix over a field. If the matrix $A$ is invertible, then the minimal polynomial $m_A$ satisfies $m_A(0) \neq 0$.
There is one definition I am unsure of or need help making more clear.
I will proceed with proof by contraposition:
We must show that if $m_A(0) = 0$ then $A$ is not invertible. By definition of minimum polynomial of $A$ we have:
$m_A(x) = x^r - \lambda_{r-1} x^{r-1} - \ldots - \lambda_1 x + \det(A)$. Not sure about the determinant term here
So, $m_A(0) = \det(A) = 0$. We know $\det(A) = 0 \implies A$ is not invertible.