Here is the problem:
Determine which of the following statements are true for all invertible $n \times n$ matrices $A$ and $B$:
A. $(A+A^{−1})^7=A^7+A^{−7}$
B. $(I_n−A)(I_n+A)=In−A^2$
C. $AB=BA$
D. $A^6B^5$ is invertible
E. $A+I_n$ is invertible
F. $(A+B)(A−B)=A^2−B^2$
So far, I believe that all except choice B are false. However, only selecting choice B as true is incorrect.
Here is my rationale for each statement:
A is incorrect as I have found a counter example with the $2\times 2$ matrix $\begin{bmatrix}1 &2 \\ 3 & 4\end{bmatrix}$.
C is incorrect as matrix multiplication is not commutative
D is incorrect as matrix multiplication must occur with matrices with the same column as row number of matrix $A$ and $B$ respectfully
E is incorrect as if $A$ were the $2 \times 2$ matrix $\begin{bmatrix}-1 & 0\\ 0 & -1\end{bmatrix}$ the resulting matrix could not be invertible
F is incorrect by the same logic as my explanation for C
I'd appreciate any guidance on this problem!
Best Answer
D) is true as already pointed out. For A) I would use the simpler example of diagonal matrix $A$ with diagonal entries $(1,0,..,0)$.
For C) you cannot just say matrix multiplication is not commutative. You have to give specific example of invertible matrices $A$ and $B$ with $AB \neq BA$. I will leave that to you. This would also answer F).
B) is true.
For E) take $A=-I_n$.