proving the following statements?

Question

1. if B is invertible then AB^-1 = B^-1A if and only if AB = BA

2. prove that if A is invertible and B is not invertible then AB is not invertible

thanks for the help

Answer 1

suppose AB = BA let B¬ denote inverse
then A = ABB¬ = BAB¬
so B¬A=B¬BAB¬= AB¬

supppose AB¬=B¬A
AB¬ = B¬B B¬A = B¬A

so AB¬=B¬A iff AB=BA

suppose A and AB are invertible with inverses A¬ and C
ABC = 1
A¬ABC = A¬
BC = A¬
BCA = A¬A = 1
i.e CA is the inverse of B
so if B is not invertible AB can't be invertible.