Let A and B be two 3 × 3 real matrices such that ( A ^ 2 - B ^ 2 )...

Let $C = {A}^{2} - {B}^{2}; \left|C\right| \ne 0$

and ${A}^{5} = {B}^{5} . . . \left(1\right)$

${A}^{3}{B}^{2} = {A}^{2}{B}^{3} . . . \left(2\right)</p><p>Subtracting equation \left(2\right)$ from \left(1\right), we get${A}^{5} - {A}^{3}{B}^{2} = {B}^{5} - {A}^{2}{B}^{3}$

$⇒ {A}^{3}\left({A}^{2} - {B}^{2}\right)$ + {B}^{3}\left({A}^{2} - {B}^{2}\right)$= O$

Post multiplying inverse of ${A}^{2} - {B}^{2}:$

${A}^{3} + {B}^{3} = O ⇒ \left|{A}^{3} + {B}^{3}\right|$ = 0$</p>$