How to calculate eigenvalues and matrix A from these:

Question

If I have matrix $A$ and I know eigenvectors $\mathbf v_1$ and $\mathbf v_2$:

a) How can I calculate eigenvalues for vectors $\mathbf v_1$ and $\mathbf v_2$

b) How can I solve $a,b,c,d$

Can you give me steps what do calculate to get right answers?

Answer 1

You know $A v_1 = \lambda_1 v_1$ and $Av_2 = \lambda_2 v_2$ where $\lambda_1$ is the eigenvalue corresponding to $v_1$ and $\lambda_2$ is the eigenvalue corresponding to $v_2$. Looking at the third component, you can calculate $\lambda_1$ and $\lambda_2$.

Now you get a system of linear equations for $a,b,c,d$ which you can solve.