Question

Let $M$ and $N$ be two $3\times 3$ matrices such that $MN=NM.$ Further, if $M\ne N^2$ and $M^2=N^4,$ then

• A. determinant of $(M^2+M{N}^2)$ is $0$
• B. there is a $3\times 3$ non-zero matrix $U$ such that $(M^2+M{N}^2)U$ is the zero matrix
• C. determinant of $(M^2+M{N}^2)\ge 1$
• D. for a $3\times 3$ matrix $U,$ if $(M^2+M{N}^2)U$ equals the zero matrix then $U$ is the zero matrix

Answer 1

Answer: (A), (B)

there is a $3\times 3$ non-zero matrix $U$ such that $(M^2+M{N}^2)U$ is the zero matrix

Given :
$MN=NM,M\ne N^2$
and $M^2=N^4$
$\Rightarrow M^2-N^4=O\Rightarrow (M-N^2)(M+N^2)=O(\because MN=NM)$

Case I :
$(M+N^2)=O\Rightarrow |M+N^2|=0$

Case II :
$|M+N^2|=0,|M-N^2|=0$
So in both the cases, $|M+N^2|=0$
Therefore,
$|(M^2+M{N}^2)|=|M||M+N^2|=0$

$(M^2+M{N}^2)U=O$
As $|M+N^2|=0$ so infinite solutions is possible.
So $|M^2+M{N}^2|\ge 1$ is not correct
also if $(M^2+M{N}^2)U=0$ then $U$ can be zero but not a must.