Let *A*, *B* and *C* be n×*n* matrices. Which of the following statements are true?

If *A* and *B* commute with *C*, then *A* commutes with B.

If *A* commutes with *B*, then *B* commutes with *A*.

If *A* commutes with B and *C*, then *A* commutes with BC.

If *A* commutes with *B*, and *B* commutes with *C*, then *A* commutes with *C*.

Let's analyze each statement regarding the commutativity of matrices to determine which are true.

### Statement 1: 
**If \( A \) and \( B \) commute with \( C \), then \( A \) commutes with \( B \).**

This statement suggests that if both \( A \) and \( B \) commute with a common matrix \( C \) (i.e., \( AC = CA \) and \( BC = CB \)), then \( A \) and \( B \) commute with each other (i.e., \( AB = BA \)).

- This is **not necessarily true**. Commuting with the same matrix \( C \) does not guarantee that two matrices \( A \) and \( B \) commute with each other. For example, in many cases, matrices \( A \) and \( B \) can commute with the identity matrix (which commutes with everything) but not commute with each other.

**Conclusion**: This statement is **false**.

### Statement 2: 
**If \( A \) commutes with \( B \), then \( B \) commutes with \( A \).**

This statement is asking whether commutativity is symmetric. In other words, if \( AB = BA \), then does \( BA = AB \)?

- This is **true** by definition. If \( A \) commutes with \( B \), then the reverse also holds by symmetry of matrix multiplication.

**Conclusion**: This statement is **true**.

### Statement 3: 
**If \( A \) commutes with \( B \) and \( C \), then \( A \) commutes with \( BC \).**

This statement suggests that if \( A \) commutes with both \( B \) and \( C \) individually (i.e., \( AB = BA \) and \( AC = CA \)), then \( A \) will commute with the product \( BC \) (i.e., \( A(BC) = (BC)A \)).

- This is **true**. If \( A \) commutes with both \( B \) and \( C \), then:
  \[
  A(BC) = A \cdot B \cdot C = B \cdot A \cdot C = B \cdot C \cdot A = (BC)A.
  \]
  Hence, \( A \) commutes with \( BC \).

**Conclusion**: This statement is **true**.

### Statement 4: 
**If \( A \) commutes with \( B \), and \( B \) commutes with \( C \), then \( A \) commutes with \( C \).**

This statement is asking whether commuting relationships are transitive. If \( A \) commutes with \( B \), and \( B \) commutes with \( C \), does that imply \( A \) commutes with \( C \)?

- This is **not necessarily true**. Commutativity is generally not transitive. There can exist matrices \( A \), \( B \), and \( C \) such that \( AB = BA \) and \( BC = CB \), but \( AC \neq CA \).

**Conclusion**: This statement is **false**.

### Final Answer:
- Statement 1: **False**
- Statement 2: **True**
- Statement 3: **True**
- Statement 4: **False**

Would you like further details or have any questions about this? Here are some related questions to explore further:

1. Can you find an example where two matrices commute with a third matrix but not with each other?
2. How does commutativity apply in higher dimensions or with operators beyond matrices?
3. What are some real-world applications of commutative matrices in physics or engineering?
4. Is it possible for non-square matrices to have commutative properties in any sense?
5. How does commutativity change when working with block matrices or tensors?

**Tip**: When two matrices commute, they often share the same set of eigenvectors, which can simplify many problems, especially in quantum mechanics or diagonalization tasks.

Let A, B and C be n×n matrices. Which of the following statements are true?

1. If A and B commute with C, then A commutes with B.
2. If A commutes with B, then B commutes with A.
3. If A commutes with B and C, then A commutes with BC.
4. If A commutes with B, and B commutes with C, then A commutes with C.

Explore true and false statements about matrix commutativity, such as whether commuting with one matrix guarantees commutativity with another. This problem involves key concepts from linear algebra, matrix theory, and commutativity properties. Ideal for undergraduate mathematics students.

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