If $v$ is an eigenvalue of $A$ and $c$ is an eigenvalue of $B$, must $vc$ be an eigenvalue of $AB$? Let $v$ be an eigenvalue of $A$ and $c$ be an eigenvalue of $B$. Is the product of $v$ and $c$ equal to an eigenvalue of $AB$?

$1$ is an eigenvalue of $
        \begin{pmatrix}
        1 & 0\\
        0 & 2\\
        \end{pmatrix},$
$3$ is an eigenvalue of $
        \begin{pmatrix}
        2 & 0\\
        0 & 3\\
        \end{pmatrix}$ 
but $3$ is not an eigenvalue of $
        \begin{pmatrix}
        1 & 0\\
        0 & 2\\
        \end{pmatrix}
        \begin{pmatrix}
        2 & 0\\
        0 & 3\\
        \end{pmatrix}=
        \begin{pmatrix}
        2 & 0\\
        0 & 6\\
        \end{pmatrix}.$
