Eigenvalues and Eigenvectors to find $A\textbf{v}$ If $A$ is a $3\times 3$ matrix with eigenvector $\begin{bmatrix} 3\\ 0\\ -2\end{bmatrix}$ corresponding to an eigenvalue of $5$ and $\begin{bmatrix} -1\\ 2\\ 7\end{bmatrix}$ corresponding to an eigenvalue of $2$, and $\textbf{v}=\begin{bmatrix}10 \\ 4 \\10\end{bmatrix}$ find $A\textbf{v}$.
My attempt at solving this problem was to begin by stating that $Ax=\lambda x$, Where $\lambda$ is the eigenvalue and $x$ is the eigenvector and then saying that $$A=5*(3,0,-2)+2*(-1,2,7)=(13,4,4)=A.$$ Then I followed by saying that $Av=(13,4,4)*(10,4,10)=186$ but I'm sure If I'm moving in the right direction.. 
 A: $Ax = λx \Leftrightarrow (Α-λ)x = 0$
Let :
$ A =  \left( \begin{array}{ccc}
a_1 & a_2 & a_3 \\
a_4 & a_5 & a_6 \\
a_7 & a_8 & a_9 \end{array} \right) $ 
Then, $ \left( \begin{array}{ccc}
a_1 & a_2 & a_3 \\
a_4 & a_5 & a_6 \\
a_7 & a_8 & a_9 \end{array} \right)
\left( \begin{array}{ccc}
3  \\
0  \\
-2  \end{array} \right) = 5 \left( \begin{array}{ccc}
3  \\
0  \\
-2  \end{array} \right)  $ 
and 
$ \left( \begin{array}{ccc}
a_1 & a_2 & a_3 \\
a_4 & a_5 & a_6 \\
a_7 & a_8 & a_9 \end{array} \right)
\left( \begin{array}{ccc}
-1 \\
2 \\
7  \end{array} \right) = 2 \left( \begin{array}{ccc}
-1  \\
2  \\
7  \end{array} \right)  $
Proceed first of all by solving the system that is created by the matrix equations above and you should be able to find the matrix $A$. After finding $A$, do the multiplication $Av$, where $v =[10,4,10]^T$ and find the solution to your exercise.
A: This isn't a real answer, just a sketch for the answer.
May A be 3 x 3 matrix
$\begin{bmatrix}
a_{11} & a_{12} & a_{13}  \\
a_{21} & a_{22} & a_{23}  \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}$
With
$Av = \lambda_1 v_1 \ , \ Av = \lambda_2 v_2$
I would solve those first and then proceed like you did before.
