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**Question 10 (16 pts):**
For matrices \( A \) and \( B \), where \( AB = BA \) (i.e., \( A \) and \( B \) commute), prove or disprove the following statement:

\[ (AB)^{10} = A^{10}B^{10}. \]

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Let's go through the proof or disproof of the given statement.

### Proof/Disproof:

Given that \( A \) and \( B \) are matrices and \( AB = BA \), the goal is to determine whether \( (AB)^{10} = A^{10}B^{10} \).

First, consider the general property for any two commuting matrices \( A \) and \( B \):
\[
(AB)^n = A^nB^n \quad \text{if } AB = BA.
\]

We can prove this by induction on \( n \).

**Base Case (\( n = 1 \)):**
\[
(AB)^1 = AB = A^1B^1.
\]
The statement holds true for \( n = 1 \).

**Inductive Step:**

Assume the statement is true for some \( n = k \), i.e.,
\[
(AB)^k = A^kB^k.
\]
We need to show it holds for \( n = k + 1 \).

Consider:
\[
(AB)^{k+1} = (AB)^k \cdot AB.
\]
Using the inductive hypothesis:
\[
(AB)^{k+1} = (A^kB^k) \cdot AB.
\]
Since \( A \) and \( B \) commute, we have:
\[
(A^kB^k) \cdot AB = A^k \cdot (B^k \cdot A) \cdot B = A^k \cdot (A \cdot B^k) \cdot B = A^k \cdot A \cdot B^k \cdot B = A^{k+1} \cdot B^{k+1}.
\]

Thus:
\[
(AB)^{k+1} = A^{k+1}B^{k+1}.
\]

By induction, the statement \( (AB)^n = A^nB^n \) holds for all positive integers \( n \).

**Conclusion:**

Since the above holds true for any integer \( n \), specifically for \( n = 10 \):
\[
(AB)^{10} = A^{10}B^{10}.
\]
Thus, the statement \( (AB)^{10} = A^{10}B^{10} \) is **proven to be true**.

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Would you like further details or have any specific questions? 

Here are some related questions:

1. What other properties of commuting matrices can be derived?
2. How does this proof change if the matrices \( A \) and \( B \) do not commute?
3. What are some examples of matrices where \( AB \neq BA \)?
4. How would this proof look in the case of \( (AB)^n = A^nB^n \) for \( n = 2 \)?
5. Can we generalize this result to more than two matrices?

**Tip:** When working with matrices, always check if they commute, as many matrix identities and simplifications depend on this property.

For matrices A and B, where AB = BA (i.e., A and B commute), prove or disprove the following statement: (AB)^{10} = A^{10}B^{10}.

Explore the proof of the statement (AB)^{10} = A^{10}B^{10} for commuting matrices A and B. This problem delves into key concepts in matrix algebra, such as the commutative property, and is suitable for undergraduate students studying advanced linear algebra.

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