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


A. (𝐴+𝐴−1)3=𝐴3+𝐴−3

B. (𝐴+𝐵)(𝐴−𝐵)=𝐴2−𝐵2

C. 𝐴𝐵𝐴−1=𝐵

D. 𝐴+𝐼𝑛
 is invertible
E. (𝐼𝑛+𝐴)(𝐼𝑛+𝐴−1)=2𝐼𝑛+𝐴+𝐴−1

F. 𝐴4
 is invertible


Let's analyze each statement to determine which ones are true for all invertible \( n \times n \) matrices \( A \) and \( B \):

**A.** \( (A + A^{-1})^3 = A^3 + A^{-3} \)

This statement is **false**. Matrix addition and multiplication do not follow simple distribution rules like for scalars. Expanding \( (A + A^{-1})^3 \) would not result in \( A^3 + A^{-3} \). There are cross terms, so this is not true in general.

**B.** \( (A + B)(A - B) = A^2 - B^2 \)

This statement is **false**. Matrix multiplication is not commutative, meaning \( AB \neq BA \) in general. The left-hand side would expand to \( A^2 - AB + BA - B^2 \), and \( AB \) may not equal \( BA \), so the expression does not simplify to \( A^2 - B^2 \).

**C.** \( A B A^{-1} = B \)

This statement is **false** in general. The result of \( A B A^{-1} \) is not necessarily \( B \). In fact, it's a similarity transformation, and it transforms \( B \) into a conjugate form. Only when \( B \) commutes with \( A \) (i.e., \( AB = BA \)) would this hold true.

**D.** \( A + I_n \) is invertible

This statement is **false** in general. The matrix \( A + I_n \) is not necessarily invertible for all invertible matrices \( A \). For example, if \( A = -I_n \), then \( A + I_n = 0 \), which is not invertible.

**E.** \( (I_n + A)(I_n + A^{-1}) = 2I_n + A + A^{-1} \)

This statement is **true**. Expanding both sides, we get:
\[
(I_n + A)(I_n + A^{-1}) = I_n + A + A^{-1} + I_n = 2I_n + A + A^{-1}
\]
Thus, this holds for all invertible matrices \( A \).

**F.** \( A^4 \) is invertible

This statement is **true**. If \( A \) is invertible, then \( A^{-1} \) exists. Therefore, \( A^4 = A \cdot A \cdot A \cdot A \) is also invertible, since the product of invertible matrices is invertible.

### True Statements:
- **E.** \( (I_n + A)(I_n + A^{-1}) = 2I_n + A + A^{-1} \)
- **F.** \( A^4 \) is invertible

Would you like more details on any of the explanations above?

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### Follow-up questions:
1. Why doesn't matrix multiplication follow commutative rules?
2. Can you provide an example where \( A + I_n \) is not invertible?
3. What are similarity transformations in matrices?
4. What does it mean for a matrix to be invertible?
5. How do you prove that a matrix power \( A^n \) is invertible?

### Tip:
Remember that matrix multiplication is generally **not commutative**—i.e., \( AB \neq BA \) in most cases.

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

A. (𝐴 + 𝐴−1)^3 = 𝐴^3 + 𝐴−3

B. (𝐴 + 𝐵)(𝐴 − 𝐵) = 𝐴^2 − 𝐵^2

C. 𝐴𝐵𝐴−1 = 𝐵

D. 𝐴 + 𝐼𝑛 is invertible

E. (𝐼𝑛 + 𝐴)(𝐼𝑛 + 𝐴−1) = 2𝐼𝑛 + 𝐴 + 𝐴−1

F. 𝐴^4 is invertible

Analyze true and false statements regarding invertible 𝑛×𝑛 matrices 𝐴 and 𝐵. Explore matrix algebra concepts, including matrix multiplication, inverses, and non-commutativity. Suitable for undergraduate students studying linear algebra.

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