Key Concept: Binomial Theorem for Matrices

The core idea to efficiently calculate powers of the given matrix $P$ is to express it as the sum of the identity matrix $I$ and a nilpotent matrix $X$. For matrices $A$ and $B$ where $AB = BA$ (i.e., they commute), the binomial theorem states $(A+B)^n = \sum_{k=0}^n \binom{n}{k} A^{n-k} B^k$. Since the identity matrix $I$ commutes with any matrix $X$ ($IX = XI = X$), we can use this theorem for $(I+X)^n$. If $X$ is nilpotent, meaning $X^k = 0$ for some positive integer $k$, the series