given two eigevectors with two eigenvalues and no A, compute Av?

Question

So I'm barely into eigenvalues and eigenvectors and the first problem from the first section tells me:

let A be a 3x3 matrix such that [-3,4,1] (column vector) is an eigenvector corresponding to the eigenvalue 3 and [6,-3,2] is an eigenvector corresponding to the eigenvalue 2. Now if v = [3,1,3], compute Av.

In this section we haven't gone over diagonalization, or anything besides finding stuff from a given particular matrix A., and the characterization equation. I tried writing out a general form and solving for each value by setting the two equations of each row to each other, but that seems impossible because they all have parameters. can someone tell me the simplest method to solve this?

Answer 1

Note that $\bar v= (3,1,3)=(6,-3,2)+(-3,4,1)$.

Since matrix multiplication is linear, $$A\bar v=A(6,-3,2) +A(-3,4,1)=2(6,-3,2)+3(-3,4,1)=(3,6,7)$$

Answer 2

What does it mean for $u = (6,-3, 2)$ to be an eigenvector with eigenvalue 2? That $Au = 2u$. Let $w = (-3,4,1)$. We know that $Aw = 3w$. Now importantly note that $v = u + w$. Can you compute $Av$ now?