Matrix proof using determinants?

Question

If A and B are nxn matrices, AB = -BA, and n is odd, show that either A or B has no inverse

Answer 1

AB=-BA
det (AB)=det (-BA)
det (A) * det (B)=(-1)^n * det (B) * det (A)
Since n is odd, (-1)^n = -1, so
det (A) * det (B)=-det (B) * det (A)

Now, suppose that det (A)≠0 and det (B)≠0. Then we may divide both sides of this equation by det (A) * det (B) to obtain 1=-1. But 1≠-1, so our assumption that det (A)≠0 and det (B)≠0 is false, and at least one of these determinants must be 0, so at least one of A or B must be non-invertible. Q.E.D.