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.