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)

The correct options are
A There is a 3 $\times$ 3 non-zero matrix U such that $(M^2+M{N}^2)$U is the zero matrix
B Determinant of $(M^2+M{N}^2)$ is $0$

$M^2=N^4\Rightarrow M^2-N^4=0\Rightarrow (M-N^2)(M+N^2)=O$

As M, N commute.

Also, $M\ne N^2,Det\left(\right(M-N^2\left)\right(M+N^2\left)\right)=0$

As $M-N^2$ is not null $\Rightarrow Det(M+N^2)=0$

Also $Det(M^2+M{N}^2)=\left(DetM\right)\left(Det\right(M+N^2\left)\right)=0$

$\Rightarrow$ There exist non-null U such that $(M^2+M{N}^2)U=O$