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.