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Let's find \( A^7 \) given that \( A^2 = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \) and \( A^3 = \begin{pmatrix} 3 & 2 \\ 2 & 1 \end{pmatrix} \).

### Step 1: Identify the Pattern or Use the Recurrence Relation

To find \( A^7 \), we can use the Cayley-Hamilton theorem or look for a recurrence relation involving the powers of \( A \).

Let's denote \( A^n \) as the matrix raised to the power \( n \).

Given:
\[
A^2 = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}, \quad A^3 = \begin{pmatrix} 3 & 2 \\ 2 & 1 \end{pmatrix}
\]

### Step 2: Express Higher Powers Using the Given Matrices

We can find the matrices \( A^4 \) and so on by multiplying the known powers:

\[
A^4 = A^3 \cdot A = \begin{pmatrix} 3 & 2 \\ 2 & 1 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}
\]

Perform the matrix multiplication:

\[
A^4 = \begin{pmatrix} 3(2) + 2(1) & 3(1) + 2(1) \\ 2(2) + 1(1) & 2(1) + 1(1) \end{pmatrix} = \begin{pmatrix} 8 & 5 \\ 5 & 3 \end{pmatrix}
\]

Continue with:

\[
A^5 = A^4 \cdot A = \begin{pmatrix} 8 & 5 \\ 5 & 3 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}
\]

\[
A^5 = \begin{pmatrix} 8(2) + 5(1) & 8(1) + 5(1) \\ 5(2) + 3(1) & 5(1) + 3(1) \end{pmatrix} = \begin{pmatrix} 21 & 13 \\ 13 & 8 \end{pmatrix}
\]

And similarly:

\[
A^6 = A^5 \cdot A = \begin{pmatrix} 21 & 13 \\ 13 & 8 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}
\]

\[
A^6 = \begin{pmatrix} 21(2) + 13(1) & 21(1) + 13(1) \\ 13(2) + 8(1) & 13(1) + 8(1) \end{pmatrix} = \begin{pmatrix} 55 & 34 \\ 34 & 21 \end{pmatrix}
\]

Finally, for \( A^7 \):

\[
A^7 = A^6 \cdot A = \begin{pmatrix} 55 & 34 \\ 34 & 21 \end{pmatrix} \cdot \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}
\]

\[
A^7 = \begin{pmatrix} 55(2) + 34(1) & 55(1) + 34(1) \\ 34(2) + 21(1) & 34(1) + 21(1) \end{pmatrix} = \begin{pmatrix} 144 & 89 \\ 89 & 55 \end{pmatrix}
\]

### Final Result

Thus, the matrix \( A^7 \) is:
\[
\boxed{A^7 = \begin{pmatrix} 144 & 89 \\ 89 & 55 \end{pmatrix}}
\]

If you have any questions or need further clarification, feel free to ask!

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Here are 5 related questions to explore:

1. How would you find \( A^8 \) given the result for \( A^7 \)?
2. What general pattern can you observe from the powers of matrix \( A \)?
3. How can you express \( A^n \) for any \( n \) using a closed-form formula?
4. What are the eigenvalues of matrix \( A \) given the patterns in its powers?
5. How does the Fibonacci sequence relate to the powers of matrix \( A \)?

**

Explore how to find the seventh power of matrix A step-by-step. Understand the use of matrix algebra, including matrix multiplication and the Cayley-Hamilton theorem. Suitable for advanced mathematics students.

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