3) (15 points)
The matrix $A=\left[\begin{array}{ll}3 & 1 \\ 2 & 2\end{array}\right]$ has eigenvectors
\[
u=\left[\begin{array}{l}
1 \\
1
\end{array}\right], \quad v=\left[\begin{array}{c}
1 \\
-2
\end{array}\right]
\]
a) (5 pts.) Determine the eigenvalue associated with $v$.
b) (5 pts.) Write $x=\left[\begin{array}{l}1 \\ 7\end{array}\right]$ as a linear combination of $u$ and $v$.
c) (5 pts.) The eigenvalue for $u$ is 4 . Use your result in (b) to determine $A^{3} x$.

Step 1 ：Given the matrix $A=\left[\begin{array}{ll}3 & 1 \\ 2 & 2\end{array}\right]$ and the eigenvector $v=\left[\begin{array}{c}1 \\ -2\end{array}\right]$, we can find the associated eigenvalue by using the property of eigenvectors and eigenvalues that $Av = \lambda v$.

Step 2 ：Multiplying the matrix $A$ by the eigenvector $v$ gives $Av = \left[\begin{array}{c}1 \\ -2\end{array}\right]$.

Step 3 ：Dividing $Av$ by $v$ gives $\lambda_v = \left[\begin{array}{c}1 \\ 1\end{array}\right]$.

Step 4 ：Since the result of the division is $\left[\begin{array}{c}1 \\ 1\end{array}\right]$, this means that the eigenvalue associated with the eigenvector $v$ is 1.

Step 5 ：Final Answer: The eigenvalue associated with the eigenvector $v$ is \(\boxed{1}\).