Matrices & Determinants
Matrix Power Recursion from P²=I-P
nta_pyq_2023_apr
Grade 12
Question:
Let $P$ be a square matrix such that $P^2=I-P$. For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$, if $P^\alpha+P^\beta=\gamma I-29P$ and $P^\alpha-P^\beta=\delta I-13P$, then $\alpha+\beta+\gamma-\delta$ is equal to
Step-by-Step Solution
Key Concept: Build powers recursively: $P^2=I-P$, $P^4=2I-3P$, $P^6=5I-8P$, $P^8=13I-21P$. Adding and subtracting.
$\alpha=8,\beta=6,\gamma=18,\delta=8$. Sum $=24$.
Correct Answer: 4