Minimal polynomial coefficient

Question

If i have a square matrix over a field and the matrix is invertible how do I show that the coefficient a0 in the minimal polynomial is Not zero?

Thx

Answer 1

Let $X^m+a_{m-1}X^{m-1}+a_1X+a_0$ is the minimal polynomial of $A$ and that $a_0=0$. then $(A^{m-1}+a_{m-1}A^{m-2}+\ldots +a_1I)\cdot A=0$, and as $A$ is invertible, $A^{m-1}+a_{m-1}A^{m-2}+\ldots +a_1I=0$, contradicting minimality.

Answer 2

The constant coeficient of the minimal polynomial is cero iff cero is root of the minimal polinomial iff cero is root of characteristic polynomial iff the matrix has cero as eigenvalue iff the matrix is singular.