Finding Eigenvalues and Eigenvectors for Leslie Matrix A Leslie Matrix is given by 
$$L =\begin{pmatrix}0 & (3/2)a^2 & (3/2)a^3\\1/2 & 0 & 0\\ 0 & 1/3 & 0\end{pmatrix}\cdot$$
Find the Eigenvalues and determine the dominant eigenvalue and eigenvector. I am struggling with how to do this without a calculator, even when I use Wolfram alpha, the answers I am getting do not seem to make sense. Should I guess a root?

The Eigenvalues are those that satisfy: $Av= \lambda v$, $(A-\lambda I_n)v=0$ where $v \neq 0$. That implies that $(A-\lambda I_n)=0$, in other words $rank(A-\lambda I_n) < n$. We can then conclude $det\{A- \lambda I_n\}=0$. You following me?
So, just solve $det\{A- \lambda I_3\}=0$ in your case by applying a Laplace expansion along column 3.  
