Suppose I have a matrix $A \in \mathbb{R}^{2, 2}$ which is the product of $3$ other matrices, lets call them $A\_1 = \left(\begin{matrix} cosx & -sinx \\ sinx & cos x\end{matrix}\right)$, $A\_2= \left(\begin{matrix} 3 & 0 \\ 0 & 7\end{matrix}\right)$ and $A\_3 = \left(\begin{matrix} cosx & sinx \\ -sinx & cos x\end{matrix}\right)$. So the situation is:
$$A = A\_1 \cdot A\_2 \cdot A\_3$$

Now, I am asked to find the eigenvalues for this matrix $A$ **without multiplying $A\_1$, $A\_2$ and $A\_3$**. 

In general, if we assume that we don't want zero eigenvectors, then we can find the eigenvalues solving the *characteristic polynomial*, which can be found by assuming (actually it is a consequence) that:
$$\det{(\lambda \cdot I - A)} = 0$$

Then we solve for $\lambda$.

Now, my situation is the following:

$$\det{(\lambda \cdot I - A\_1 \cdot A\_2 \cdot A\_3)} = 0$$

I have found the determinant of $A$, using a property of the determinants, but I don't see any property that can help me to find the determinant just above...