How do i prove that $\det(tI-A)$ is a polynomial?

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).