Putnam 1991 A2

Let ${\mathbf{A}}$ and ${\mathbf{B}}$ be different $n \times n$ matrices with real entries. If ${\mathbf{A}}^3 = {\mathbf{B}}^3$ and ${\mathbf{A}}^2 {\mathbf{B}} = {\mathbf{B}}^2 {\mathbf{A}}$, can ${\mathbf{A}}^2 + {\mathbf{B}}^2$ be invertible?