Exercise 5.1.18

a nonzero polynomial of $c$. It has only finite zeroes, so we can always find some $c$ sucht that the determinant is nonzero.

take $A=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$ and $B=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$.