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I am just wondering what the answer to this question is. I have been trying it for 4 days now and i can't seem to get the right answer.

Determine which of the formulas hold for all invertible nxn matrices A and B

A. (A+B)^2 = A^2+B^2+2AB

B. ABA^-1=B

C. (I_n-A)(I_n+A)=I_n-A^2

D. 8A is invertible

E. (AB)^-1=A^-1B^-1

F. A+A^-1 is invertible

I have gotten B , C , D , F as my answers but my teacher is telling me that they are not the answers. Any help would be greatly appreciated.

A. Not true in general since A and B don't necessarily commute.

B. Not true; need to commute.

C. True: Distributive law works for matrices and IA = AI always.

D. It looks like the number 8 is multiplying A; typo? If it is BA, then (with help from the associative law) (BA)^-1(BA) = A^-1(B^-1B)A =A^-1A = I, so that it is invertible.

E. Not true, however, (AB)^-1 = (B^-1)(A^-1).

F. Not true: try multiplying it by its inverse and seeing if it comes out equal to I.

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