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 $A$?

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