This problem delves into the properties of matrices, specifically focusing on factorization and the implications of invertibility. The core idea is to cleverly manipulate the given matrix equations to isolate the term $(A^3 + B^3)$ and utilize the information about the invertibility of $(A^2 - B^2)$.

1. Key Concept: Matrix Invertibility and Factorization

A matrix $M$ is invertible if and only if its determinant $\det(M) \ne 0$. If $M$ is invertible, then for any matrix $X$, the equation $MX = 0$ implies $X = 0$ (by multiplying by $M^{-1}$ on the left). Similarly, $XM = 0$ implies $X = 0$. This property