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 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)

$M^2=N^4$
$\Rightarrow M^2-N^4=O$
$\Rightarrow \left(M-N^2\right)\left(M+N^2\right)=O$
As M, N commute.
Also, $M\ne N^2,\text{Det}\left(\left(M-N^2\right)\left(M+N^2\right)\right)=0$
As $M-N^2$ is not null
$\Rightarrow \text{Det}\left(M+N^2\right)=0$
Also Det $\left(M^2+M{N}^2\right)=($ Det $M)($ Det $\left(M+N^2\right))=0$
$\Rightarrow$ There exist non-null $U$ such that $\left(M^2+M{N}^2\right)U=O$