TITLE: Finding the value of the determinant $|M^2+MN^2|$ and determining whether $U$ is a zero matrix or not
QUESTION [5 upvotes]: Let $M$ and $N$ be two $3\times 3$ matrices such that $MN=NM$. Further
  if $M\neq N^2$ and $M^2=N^4$ then what is/are the possible value(s) of
  the determinant $|M^2+MN^2|$? Also if there is a $3\times 3$ matrix
  $U$ such that $(M^2+MN^2)U=O$ can we say whether $U$ is a zero matrix or
  a non-zero matrix ?


My Attempt:
$|M|=\pm|N|^2$ (from $M^2=N^4$)

$(M-N)(N-M)=M^2-N^2$ (as $MN=NM$)

$M^2=N^4$ 
$\implies MMN=N^5$ 
$\implies NMMN=N^6$ 
$\implies MNMN=N^6$ 
$\implies (MN)^2=N^6$

Similarly, $M^6=(MN)^4$

These are what I could deduce from the given information. How should I proceed from here to solve the question?

REPLY [3 votes]: Let's first note that $M$ and $N^2$ commute because $M$ and $N$ do. Because of this, we may write $$0 = M^2 - N^4 = (M-N^2)(M+N^2).$$
This tells us that $|M+N^2| = 0$, since if it wasn't, you could multiply both sides by the inverse $(M+N^2)^{-1}$ to find $0 = M - N^2$ and hence $M = N^2$, which isn't allowed. 
But $M^2 + MN^2 = M(M+N^2)$, hence $$|M(M+N^2)| = |M||M+N^2| = |M| \cdot 0 = 0.$$
As for the second part, you can't say anything about $U$. For example, take $M = -I$ and $N = I$. These satisfy the conditions, but $M^2 + MN^2 = 0$, so obviously $(M^2 + MN^2)U = 0$ for any matrix $U$.