Let M and N be two 3 × 3 matrices such that MN=NM, further, if $\mathrm{M}\ne {\mathrm{N}}^2\text{ and }{\mathrm{M}}^2={\mathrm{N}}^4$then

(i) If A and B are two non-zero matrices and AB=BA, then (A–B)(A+B)=A²–B^2.
(ii) The determinant of the product of the matrices is equal to product of their individual determinants i.e $\left|\mathrm{AB}\right|=\left|\mathrm{A}\right|\left|\mathrm{B}\right|$
Given, ${\mathrm{M}}^2={\mathrm{N}}^4\Rightarrow {\mathrm{M}}^2-{\mathrm{N}}^4=0$ $\Rightarrow \left(\mathrm{M}-{\mathrm{N}}^2\right)\left(\mathrm{M}+{\mathrm{N}}^2\right)=0(\mathrm{as}\mathrm{MN}=\mathrm{NM})$
Also, $\mathrm{M}\ne {\mathrm{N}}^2\Rightarrow \mathrm{M}+{\mathrm{N}}^2=0\Rightarrow \mathrm{Det}\left(\mathrm{M}+{\mathrm{N}}^2\right)=0$
Also, $\mathrm{Det}\left({\mathrm{M}}^2+{\mathrm{MN}}^2\right)=(\mathrm{Det}\mathrm{M})\left(\mathrm{Det}\left(\mathrm{M}+{\mathrm{N}}^2\right)\right)$
$=(\mathrm{Det} M)\left(0\right)=0\mathrm{As},\mathrm{Det}\left({\mathrm{M}}^2+{\mathrm{MN}}^2\right)=0$