TITLE: Which of the following statements is not necessarily true?
QUESTION [6 upvotes]: Let $A∈M_{5×6}(\mathbb R)$ and $B∈M_{6×5}(\mathbb R)$ such that $(AB)^{100}=0$. Which of the following statements is not necessarily true?
(a) $(AB)^{4}=0$
(b) $(BA)^{6}=0$
(c) $\det(BA)=0$
(d) $\operatorname{tr}(AB)=0$

REPLY [8 votes]: (a) need not be true since $$A=\left( \begin{matrix} 0& 1& 0 & 0 & 0 & 0  \\
0& 0& 1 & 0 & 0 & 0 \\
0& 0& 0& 1 & 0 & 0  \\ 
0 & 0 & 0& 0& 1 &0 \\
0 & 0& 0& 0& 0& 0 \\
\end{matrix}\right) \mathrm{\qquad and \qquad} B=\left( \begin{matrix} 1& 0& 0 & 0 & 0   \\
0& 1& 0 & 0 & 0  \\
0& 0& 1& 0 & 0   \\ 
0 & 0 & 0& 1& 0  \\
0 & 0& 0& 0& 1 \\
0 & 0 & 0 & 0 & 0
\end{matrix}\right)$$
satisfy $(AB)^4\ne 0 $ but  $(AB)^5= 0 $.
Now the characteristic polynomial of $AB$ $\ $is of degree $\le 5$ and divides $X^{100} $ so it must equal $X^5$. (b) is true since $(BA)^6=B(AB)^5A=0$ by Cayley-Hamilton. (c) is true since the determinant of $AB$ is $\pm$  the constant term of the characteristic polynomial of $AB$, and (d) is true since tr$(AB)$  is $-({\mathrm{ coefficient\ of \ }}X^4)$.