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

Hint: We consider these matrices as normal determinant form as the matrices are commutative. We try to find the determinant value of $\left( {{M}^{2}}+M{{N}^{2}} \right)$. We also use a null matrix U to find $\left( {{M}^{2}}+M{{N}^{2}} \right)U$ being a zero matrix.

Complete step by step answer:
We use matrix operations to find out the correct options.
It’s given that ${{M}^{2}}={{N}^{4}}$ and $M\ne {{N}^{2}}$.
We can say that M and N are commutative as $MN=NM$.
We can consider them as multiplication form ${{M}^{2}}={{N}^{4}}\Rightarrow \left( M-{{N}^{2}} \right)\left( M+{{N}^{2}} \right)=0$.
As we know $M\ne {{N}^{2}}$, we can say that
$\begin{align}
  & \left( M-{{N}^{2}} \right)\left( M+{{N}^{2}} \right)=0 \\
 & \Rightarrow \left( M+{{N}^{2}} \right)=0 \\
\end{align}$
We have to consider them as matrix values. So, we need to use them as a determinant form.
$\det \left[ \left( M-{{N}^{2}} \right)\left( M+{{N}^{2}} \right) \right]=0$. We have $\det \left( M-{{N}^{2}} \right)\ne 0$.
This gave us $\det \left( M+{{N}^{2}} \right)=0$.
We need to find $\det \left( {{M}^{2}}+M{{N}^{2}} \right)$.
As the matrices are commutative $\det \left( {{M}^{2}}+M{{N}^{2}} \right)=\left[ \det \left( M \right) \right]\left[ \det \left( M+{{N}^{2}} \right) \right]$.
Now $\det \left( M+{{N}^{2}} \right)=0$ which gave us
$\det \left( {{M}^{2}}+M{{N}^{2}} \right)=\left[ \det \left( M \right) \right]\left[ \det \left( M+{{N}^{2}} \right) \right]=0$.
So, for any $3\times 3$ matrix U we can find $\left( {{M}^{2}}+M{{N}^{2}} \right)U$ is the zero matrix. $\left( {{M}^{2}}+M{{N}^{2}} \right)U=O$.

So, the correct answer is “Option A and B”.

Note: Considering the matrices as their determinant form and using normal binary operation is only working as the matrices are commutative. Without this condition this wouldn’t have worked.