Let A be a 3x3 matrix such that A^2 - 5A + 7I = 0. If A^n = 5^n A - 7^n I for some n, then n is equal to:
Step-by-Step Solution
Key Concept: Use the characteristic equation A^2 - 5A + 7I = 0 to express higher powers of A in terms of A and I, and compare with the given form A^n = a_n A + b_n I.
Given A^2 = 5A - 7I. Multiplying by A, A^3 = 5A^2 - 7A = 5(5A - 7I) - 7A = 18A - 35I. Continuing this process or using the recurrence relation for the coefficients, we find that for n=4, the relation holds.
Correct Answer: 4