Question

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

• A. determinant of $(M^2 + MN^2)$ is $0$
• B. there is a $3 \times 3$ non-zero matrix U such that $(M^2 + MN^2)$ U is zero matrix
• C. determinant of $(M^2 + MN^2) \ge 1$
• D. for a $3 \times 3$ matrix U, if $(M^2 + MN^2)$ 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, $M \neq N^{2}, \text{Det}\left(\left(M-N^{2}\right)\left(M+N^{2}\right)\right)=0$
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 $M)\left(\right.$ Det $\left.\left(M+N^{2}\right)\right)=0$
$\Rightarrow $ There exist non-null $U$ such that $\left( M ^{2}+ MN ^{2}\right) U = O$