if $\lambda$ is an eigenvalue of A and an eigenvalue of B is it also an eigenvalue of AB? The question is essentially explained in the title: given that $\lambda$ is an eigenvalue of an $n \times n$ matrix A and another $n \times n$ matrix B, is also an eigenvalue of AB?

No; consider the matrices $A=B=\begin{pmatrix}2&0\\\0&2\end{pmatrix}$, or more generally $A=B=\lambda I$ with $\lambda\
eq0,1$.