How do i prove that $\det(tI-A)$ is a polynomial? In wikipedia, it's said "$\det(tI-A)$ can be explicitly evaluated using exterior algebra", but i have not learned exterior algebra yet and i just want to know whether it is polynomial, not how it looks like.
How do i prove that $\det(tI-A)$ is a polynomial in $\mathbb{F}[t]$ where $\mathbb{F}$ is a field and $A$ is an $n\times n$ matrix?

To prove that $\det (tI-A)$ is a polynomial you must know some definition, or some properties, of the determinant. The most straightforward and least mystical approach is to use Laplace's formula:  http://en.wikipedia.org/wiki/Laplace_expansion
This would give a rather quick way of proving that $\det (tI-A)$ is a polynomial, and for almost the same amount of work, that it is a monic polynomial of degree $n$ (if $A$ is an $n\times n$ matrix). 
