I am really having trouble trying to see if this correct. Could you help? Select all statements below which are true for all invertible $n\times n$ matrices $A$ and $B$
A. $(A+A^{−1})^4=A^4+A^{−4}$ (true)
B. $A+I_n$ is invertible (False)
C. $A^7 B^2$ is invertible (true)
D. $AB=BA$ (False)
E. $(A+B)^2=A^2+B^2+2AB$ (False)
F. $(I_n−A)(I_n+A)=I_n−A^2$ (true)
 A: Statement A is false. The inverse of a matrix $A$ is defined such that $A^{-1}A=AA^{-1}=I$, where $I$ is the identity matrix. We can see that $(A+A^{-1})^4=A^4+A^{-4})$ by simply expanding $(A+A^{-1})^4$ using the binomial theorem. Doing so and using the above identity, we have $(A+A^{-1})^4 = A^4 + 4(A^2+A^{-2})+6I+A^{-4}\neq A^4+A^{-4}$. Take the $2\times 2$ invertible matrix $M=
  \left[ {\begin{array}{}
   2 & 0\\
   0 & 2\\
  \end{array} } \right]
$ as a counterexample.
Statement B is false. Use the matrix $-M$ and choose $n=2$. The resulting determinant will be $0$.
Statement C is true. Simply take the matrix $B^{-2}A^{-7}$ and by the properties of matrix inverses and the identity matrix, you'll get the identity matrix.
Statement D is false. Matrix multiplication is known not to be commutative and it is easy to see this if you take any two assymmetric matrices (i.e. it is not equal to its transpose).
Statement E is also false. Using the fact that statement D is false and performing the specified multiplication, we obtain $(A+B)^2 = A^2 + AB + BA + B^2 \neq A^2 + 2AB + B^2$ because it is not true that $AB= BA$ for all $n\times n$ matrices $A$ and $B$.
Statement F is false. It should read $(In-A)(In+A)=In^2-A^2$ since $n$ is an integer. This is due to the properties of matrix multiplication and the fact that $I^2 = I$.
A: $(A+A^{-1})^4 = A^4 + 4A^2 + 6I + 4A^{-2}+ A^{-4}$
$(nI + A)(nI-A) = n^2I - A^2$
