Question

Let I be an identity matrix of order 2 × 2 and P = [2 -1; 5 -3]. Then the value of n ∈ N for which Pⁿ = 5I - 8P is equal to ________.

Hint:

Cayley-Hamilton Theorem is the fastest way to find a relation for $P^n$ by expressing high powers in terms of $P$ and $I$.

Answer 1

Correct Answer: 6

Step 1: Find the characteristic equation of $P$: $|P - \lambda I| = 0$. $(2-\lambda)(-3-\lambda) - (-5) = 0 \implies \lambda^2 + \lambda - 1 = 0$. By Cayley-Hamilton Theorem: $P^2 + P - I = 0 \implies P^2 = I - P$.
Step 2: Calculate higher powers: $P^4 = (I - P)^2 = I - 2P + P^2 = I - 2P + (I - P) = 2I - 3P$.
Step 3: Calculate $P^6$: $P^6 = P^4 \cdot P^2 = (2I - 3P)(I - P) = 2I - 2P - 3P + 3P^2$ $P^6 = 2I - 5P + 3(I - P) = 2I - 5P + 3I - 3P = 5I - 8P$.
Step 4: Comparing with $P^n = 5I - 8P$, we get $n=6$.