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**lllll** Given $\displaystyle A$, an $\displaystyle n \times n$ matrix, with the following characteristic polynomial

$\displaystyle f(t) = (-1)^n t^n +a_{n-1}t^{n-1}+ \ \dotso \ + a_1t+a_0$ show that $\displaystyle A$ is invertible if only if $\displaystyle a_0\neq 0$

what I have is f(t) can be written as $\displaystyle det(A-t \cdot I)$ where $\displaystyle I$ is the identity matrix, so I would have:

$\displaystyle det \left( \begin{bmatrix}

0 & 0 & \dotso & 0 & -a_0 \\

1 & 0 & \dotso & 0 & -a_1\\

0 & 1 & \dotso & 0 & -a_2\\

\vdots & \vdots & \ddots & \vdots & \vdots\\

0 & 0 & \dotso & 1 & -a_{n-1}

\end{bmatrix} - t\times \begin{bmatrix}

1 & 0 & \dotso & 0 & 0 \\

0 & 1 & \dotso & 0 & 0\\

0 & 0 & \dotso & 0 & 0\\

\vdots & \vdots & \ddots & \vdots & \vdots\\

0 & 0 & \dotso & 0 & 1

\end{bmatrix}\right)$

now from the definition of invertible there must exist a matrix $\displaystyle B$ which is $\displaystyle n \times n$ matrix such that $\displaystyle A\cdot B = I$

but how do I find such a matrix, and what is the connections with $\displaystyle a_0$, beyond the fact that it can't be 0?