Select all statements below which are true for all invertible n×n matrices AA and BB

a. (A+A^-1)^3=A^3+A^-1
b. ABA^-1=B
c. A^8 is invertible
d. A+In is invertible
e. (In+A)(In+A^-1)=2In+A+A^-1
f. (A+B)(A−B)=A^2−B^2

(a) is false. Consider the identity matrix, . Then

bu

(b) is false. Since is invertible, we have

but in general, matrix multiplication is non-commutative.

(c) is true. Let ; then

and so on, down to

which is to say .

(d) is false. Consider , so that

which is singular (i.e. determinant is zero), and thus not invertible.

(e) is true. It follows from the distributivity of matrix multiplication:

(f) is false for the same reason as (b). Expanding the product gives

but in general, .