TITLE: If I know $AB$, how can I calculate $BA$?
QUESTION [8 upvotes]: Let $A∈\mathscr{M}_{3×2}(\mathbb{R})$ and $B∈\mathscr{M}_{2\times3}(\mathbb{R})$ be matrices satisfying
  $AB =\begin{bmatrix}
8 &2 &−2\\
2 &5 &4\\
−2 &4& 5
\end{bmatrix}$. Calculate $BA$. (Golan, The Linear Algebra a Beginning Graduate Student Ought to Know, Exercise 426.)

Maybe it can be solved by solving a system of equations, but I think there is a shorter way since this problem was in my exam.
Thanks.

REPLY [6 votes]: $(AB)^2=9(AB)\Rightarrow(BA)^3=9(BA)^2\Rightarrow \mu_{BA}(X)\mid X^2(X-9)$
$P_{AB}(X)=X(X-9)^2\Rightarrow P_{BA}(X)=(X-9)^2\Rightarrow\mu_{BA}(X)\mid (X-9)^2$
Conclusion: $\mu_{BA}(X)=X-9$, so $BA=9I_2$.
(Here $\mu_X$, respectively $P_X$ stands for the minimal polynomial, respectively the characteristic polynomial of a square matrix $X$.)