Select all statements below which are true for all invertible n×n
matrices A
and B

A. (AB)−1=A−1B−1

B. ABA−1=B

C. A2
is invertible
D. (In−A)(In+A)=In−A2

E. (A+B)(A−B)=A2−B2

F. A+B
is invertible

Let's analyze each statement to determine whether it is true for all invertible \( n \times n \) matrices \( A \) and \( B \):

**A. \( (AB)^{-1} = A^{-1}B^{-1} \)**  
This statement is **false**. The correct formula is \( (AB)^{-1} = B^{-1}A^{-1} \), not \( A^{-1}B^{-1} \).

**B. \( ABA^{-1} = B \)**  
This statement is **true**. Since \( A \) and \( A^{-1} \) are inverse matrices, they cancel each other out, leaving \( B \).

**C. \( A^2 \) is invertible**  
This statement is **true**. If \( A \) is invertible, then \( A^2 = A \times A \) is also invertible, and its inverse is \( (A^2)^{-1} = (A^{-1})^2 \).

**D. \( (I_n - A)(I_n + A) = I_n - A^2 \)**  
This statement is **false**. Expanding \( (I_n - A)(I_n + A) \) using the distributive property gives \( I_n^2 - A^2 = I_n - A^2 \), which is indeed true, but it's valid regardless of whether \( A \) is invertible. However, this equation must hold specifically for invertible matrices, and the matrix \( I_n \) has nothing to do with invertibility of \( A \).

**E. \( (A + B)(A - B) = A^2 - B^2 \)**  
This statement is **false**. For matrices, \( (A + B)(A - B) = A^2 - AB + BA - B^2 \), not simply \( A^2 - B^2 \). This only equals \( A^2 - B^2 \) if \( AB = BA \), which is not true for all matrices unless \( A \) and \( B \) commute.

**F. \( A + B \) is invertible**  
This statement is **false**. The sum \( A + B \) is not necessarily invertible. For example, if \( A = -B \), then \( A + B = 0 \), which is not invertible.

### Summary:
The only statements that are true for all invertible \( n \times n \) matrices \( A \) and \( B \) are:

- **B. \( ABA^{-1} = B \)**
- **C. \( A^2 \) is invertible**

Would you like further details or have any questions?

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Here are 5 relative questions:

1. What is the general formula for the inverse of a product of two matrices?
2. Can you provide an example where \( A + B \) is not invertible even though \( A \) and \( B \) are invertible?
3. How does the invertibility of a matrix affect the invertibility of its powers, like \( A^3 \) or \( A^4 \)?
4. Under what conditions does \( (A + B)(A - B) = A^2 - B^2 \) hold true for matrices?
5. What are the implications of the commutativity of matrices \( A \) and \( B \) on matrix multiplication?

**Tip:** Remember, matrix multiplication is not commutative, so the order of multiplication matters significantly in matrix operations.

Explore common statements about invertible matrices and their validity. Learn about matrix operations and theorems related to invertibility. Ideal for advanced high school and college students studying linear algebra.

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