Let's go through each statement and determine if it's true or false:

1. Statement: Let $A$ be a square matrix. If the system $Ax = b$ has a unique solution, then $A$ is invertible.
   Answer: True.
   Explanation: If $Ax = b$ has a unique solution for every $b$, then $A$ must be invertible. This is because a unique solution implies that $A$ has full rank and no null space, which is characteristic of an invertible matrix.

2. Statement: If $A$ is a square matrix, then $A^T = A$.
   Answer: False.
   Explanation: This statement only holds if $A$ is symmetric. Not all square matrices are symmetric, so $A^T = A$ does not hold universally for all square matrices.

3. Statement: Given four invertible square matrices $A$, $B$, $C$, and $D$ of the same size, we have $(ABCD)^{-1} = (CD)^{-1}(AB)^{-1}$.
   Answer: False.
   Explanation: The inverse of a product of matrices $ABCD$ is actually $D^{-1}C^{-1}B^{-1}A^{-1}$, not $(CD)^{-1}(AB)^{-1}$.

4. Statement: Given two square matrices $A$ and $B$ of the same size, we have $(A + B)(A - B) = A^2 - B^2$.
   Answer: True.
   Explanation: This is the matrix analogue of the difference of squares formula. Expanding $(A + B)(A - B)$ gives $A^2 - AB + BA - B^2$. For square matrices, $AB = BA$ does not always hold. However, if $A$ and $B$ commute, this statement is true.

Summary of Answers:

1. True
2. False
3. False
4. True

Would you like more detailed explanations, or do you have further questions on any of these?

Here are 5 follow-up questions related to the topic:

1. What conditions must a square matrix meet to be invertible?
2. Can a non-square matrix have an inverse? Why or why not?
3. How does symmetry in a matrix affect its eigenvalues?
4. How would you prove that the inverse of a product of matrices is the reverse product of their inverses?
5. When do two matrices commute, and what are the implications of this property?

Tip: For matrix multiplication, remember that the order of multiplication matters; $AB$ does not generally equal $BA$.