Eigenvalue of the product of two matrices with the same eigenvalues

Question

If two $n\times n$ matrices $A$ and $B$ both have the same eigenvalue $\lambda$ does this mean their product $AB$ has eigenvalue $\lambda$

Answer 1

Hint: take $A=B$ with a diagonal matrix, say, in $M_2(K)$ or even in $M_1(K)$ having an eigenvalue $\lambda>1$.

Answer 2

I think this is a counterexample. Let $A=\begin{bmatrix}1 & 2\\ 1 &0\end{bmatrix}$ and $B=\begin{bmatrix}1& 1\\ 1&1\end{bmatrix}$. The only eigenvalue these matrices share is $2$. Their product $AB$ is $\begin{bmatrix}3 & 3\\1&1\end{bmatrix}$ which has eigenvalues $0$ and $4$.