If A has eigenvector $\mathbf{v}_1$ so that $A\mathbf{v}_1=\lambda_1\mathbf{v}_1$and B has eignenvector $\mathbf{v}_2$ so that $B\mathbf{v}_2=\lambda_2\mathbf{v}_2$, then what can you say about AB? can you say $AB\mathbf{v_3}=\lambda_3\mathbf{v}_3$? and what would be the relationship between $\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ and what would be the relationship between $\lambda_1,\lambda_2,\lambda_3$?
Edit $A,B$ are 3 by 3 matrices and $\lambda_1,\lambda_2,\lambda_3$ can be real or complex numbers and $\mathbf{v}_1,\mathbf{v}_2,\mathbf{v}_3$ is a triple.