Let $A$ be a matrix of size $5$ (real values) and real $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, such that:
vector $(1,1,1,1,1)$ is eigenvector of $A$ corresponding to eigenvalue $\lambda_{1}$
vector $(1,2,3,4,5)$ is eigenvector of $A$ corresponding to eigenvalue $\lambda_{2}$
vector $(1,3,5,7,9)$ is eigenvector of $A$ corresponding to eigenvalue $\lambda_{3}$
Prove that vector $(43,53,63,73,83)$ is eigenvector of $A$ corresponding to eigenvalue $3\lambda_1 + 5\lambda_2 - 7\lambda_3$.
I know that this set of eigenvectors is not lineary independent and I think that this might be important but I don't know how to use this information.