Show that if AB is invertible, so is B.

Question

Show that if AB is invertible, so is B.

Answer 1

Take the determinant: 0≠det(AB)=det(A)*det(B), where we've used the multiplicative property of determinants. So det(B)≠0, which means that B is invertible.

Alternative approach: suppose that Bv = 0. Then A(Bv)=(AB)v = 0. But AB is invertible, so its null space is 0, which means that v=0. Thus the null space of B is {0}, which implies that B is invertible.