https://file.aiquickdraw.com/mathbot/file/1730248441203ofinbo39.jpg

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?

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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 \( BA \).

1. Let A be a square matrix. If the system Ax = b has a unique solution, then A is invertible. True or False?
2. If A is a square matrix, then A^T = A. True or False?
3. Given four invertible square matrices A, B, C, and D of the same size, we have (ABCD)^{-1} = (CD)^{-1}(AB)^{-1}. True or False?
4. Given two square matrices A and B of the same size, we have (A + B)(A - B) = A^2 - B^2. True or False?

Analyze statements on matrix properties and determine their truth values. Topics include matrix inverses, symmetry, and matrix identities. Suitable for undergraduate students studying Linear Algebra.

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