For how many $\lambda\in\mathbb{C}, \det(\lambda E-A)=0$ Suppose the polynomial $p(\lambda)=\det(\lambda E-A)$ not identially $=0$, 
 i.e $p(\lambda)\neq 0$, for fixed $E,A\in M_n(\mathbb{R})$ and $\lambda$ is a complex variable.

Here are my questions

$1.$ Is it always true that degree of $p$ is always $n$? If $E=I_n$, then we know that it is a characteristic polynomial of $A$ and since $A$ is of $n\times n$ so $p$ is of $n$ degree polynomial, right? is it true in general for any invertible $E$? what is $E$ is singular?

$2.$ If degree of $p$ is $n$ then there are $\lambda_1,\dots,\lambda_n$ such that $p(\lambda_i)=0$ for $i=1,2,\dots ,n$

does this depends on rank of $E$? I mean if $E$ is singular and $rank(E)=n_1&lt;n$, then what can be said about the degree of $p$ and number of zeros of $p$

Thanks
It is a bit more complicated than that.  You may rewrite it in a form like:
$$ {\rm const} \times \det  \left( \begin{matrix} \lambda 1_{n_1} - A_{11} &amp; A_{12} \\ A_{21} &amp; A_{22} \end{matrix} \right) $$
where the upper left matrix has size $n_1={\rm rank}\ E$. Now the coefficient to $\lambda^{n_1}$ is $\det A_{22}$ so if this is zero there is no term with $\lambda^{n_1}$ and you will have to look at the next coefficient which also involves $A_{11}$, $A_{12}$ and $A_{21}$. I don't know if there is an easy general description of the degree.
The degree of $p(\lambda)$ is at most the rank of $E$.

To see this, note that there exist invertible matrices $U$ and $V$ such that $UEV$ is diagonal with $1$s and $0$s, and that
$$
\det(\lambda E - A)= \det(U^{-1})\det(\lambda UEV - UAV)\det(V^{-1})
$$
By the fundamental theorem of algebra, a degree $n$ polynomial has $n$ roots.