Let A and B be two 3 × 3 real matrices such that A 2 - B 2 is invertible matrix. If A 5 = B 5 and A 3…

Let $C=A^2-B^2;\left|C\right|\ne 0$

and $A^5=B^5 ...\left(1\right)$

$A^3{ B}^2=A^2{ B}^3 ...\left(2\right)$

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

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

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

$A^3+B^3=O\Rightarrow \left|A^3+B^3\right|=0$