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The problem asks to provide an example of two \(3 \times 3\) matrices \( A \) and \( B \), neither of which is the zero matrix nor the identity matrix, such that \( AB = BA \) (i.e., they commute).

A good approach to this problem is to find matrices that have special structures, such as diagonal matrices, which often commute. Let's construct two simple diagonal matrices, since diagonal matrices generally commute if they have similar diagonal entries.

### Example of commuting matrices \( A \) and \( B \):

\[
A = \begin{pmatrix}
2 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 4
\end{pmatrix}
\]

\[
B = \begin{pmatrix}
5 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 7
\end{pmatrix}
\]

These matrices satisfy the condition \( AB = BA \). Here's why:
- Since both \( A \) and \( B \) are diagonal matrices, their product is also a diagonal matrix.
- Multiplying two diagonal matrices results in another diagonal matrix where the diagonal entries are simply the product of the corresponding diagonal entries of the matrices.
- Matrix multiplication is commutative for diagonal matrices (i.e., \( AB = BA \)).

This example avoids both the identity matrix and the zero matrix, as required.

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Do you want more details on this example or have any questions? Here are 5 related questions to extend your understanding:

1. How can you prove that two specific matrices commute in general?
2. What properties of diagonal matrices ensure that they commute?
3. Can you find a pair of non-diagonal matrices that commute?
4. How does matrix commutativity relate to eigenvalues and eigenvectors?
5. What would change in the example if one matrix was not diagonal?

**Tip:** When working with matrices, always check for special properties (e.g., diagonal, triangular) that might simplify operations like multiplication.

Give an example of two 3 × 3 matrices A and B, neither of which is the zero matrix or the identity matrix, such that AB = BA.

Explore an example of two 3x3 matrices A and B that commute, meaning AB = BA, while avoiding the identity and zero matrices. This solution demonstrates the properties of diagonal matrices that enable commutative multiplication.

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