To solve for $A^9x$, we can leverage the fact that $A$ is a diagonalizable matrix. Given the eigenvectors $v_1$ and $v_2$, and their corresponding eigenvalues $\lambda_1$ and $\lambda_2$, the matrix $A$ can be represented as:

$A = PDP^{-1}$

where $P$ is the matrix of eigenvectors and $D$ is the diagonal matrix of eigenvalues.

Given:

• Eigenvectors: $v_1 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$ and $v_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$
• Eigenvalues: $\lambda_1 = \frac{1}{2}$ and $\lambda_2 = 2$
• Vector $x = \begin{pmatrix} 5 \\ -1 \end{pmatrix}$

1. Construct the matrix $P$ and $D$:

$P = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}, \quad D = \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & 2 \end{pmatrix}$

2. Find the inverse of $P$:

$P^{-1} = \frac{1}{2} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$

3. Compute $A^9$:

Since $A = PDP^{-1}$, then $A^9 = PD^9P^{-1}$.

$D^9 = \begin{pmatrix} \left(\frac{1}{2}\right)^9 & 0 \\ 0 & 2^9 \end{pmatrix} = \begin{pmatrix} \frac{1}{512} & 0 \\ 0 & 512 \end{pmatrix}$

$A^9 = P \begin{pmatrix} \frac{1}{512} & 0 \\ 0 & 512 \end{pmatrix} P^{-1}$

4. Calculate $A^9x$:

$A^9x = PD^9P^{-1}x$

First, find $P^{-1}x$:

$P^{-1}x = \frac{1}{2} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 5 \\ -1 \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 5 - (-1) \\ 5 + (-1) \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 6 \\ 4 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$

Then, multiply by $D^9$:

$D^9 \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} \frac{1}{512} & 0 \\ 0 & 512 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} \frac{3}{512} \\ 1024 \end{pmatrix}$

Finally, multiply by $P$:

$A^9x = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} \frac{3}{512} \\ 1024 \end{pmatrix} = \begin{pmatrix} \frac{3}{512} + 1024 \\ -\frac{3}{512} + 1024 \end{pmatrix} = \begin{pmatrix} 1024.005859375 \\ 1023.994140625 \end{pmatrix}$

So,

$A^9x \approx \begin{pmatrix} 1024.01 \\ 1023.99 \end{pmatrix}$

Would you like more details on any part of this process?

Related Questions:

1. How do you diagonalize a matrix?
2. What are the properties of eigenvalues and eigenvectors in matrix multiplication?
3. How do you compute matrix powers using eigenvalues and eigenvectors?
4. What are some practical applications of diagonalization in engineering?
5. How can one find the inverse of a 2x2 matrix quickly?

Tip:

When dealing with matrix exponentiation, diagonalization is a powerful tool that simplifies the computation, especially for high powers.