Let M = [ 0, - α; α, 0 ], where α is a non-zero real number an N = Σₖ₌₁⁴⁹ M²ᵏ. If (I - M²)N = - 2I, then the positive integral value of α is ____________.

Solution

Maths sir · 4 steps

Square M first: it collapses to a multiple of I, and the whole sum becomes an ordinary GP.

1. Square M. Multiplying out gives M² = (-α², 0; 0, -α²) = -α² I. Write t = -α², so M²ᵏ = tᵏ I.
2. Sum becomes a GP. Now N = (t + t² + … + t⁴⁹) I and I - M² = (1 - t) I. Their product telescopes, because (1-t)(t + t² + … + t⁴⁹) = t - t⁵⁰.

   (I - M²)N = (t - t⁵⁰) I = (-α² - α¹⁰⁰) I

3. Match with −2I. Set -α² - α¹⁰⁰ = -2, so α² + α¹⁰⁰ = 2. The left side increases with α², so it has only one solution: α² = 1.
4. So. α = ± 1, and the positive integral value is 1.

The idea to keep. When a power of a matrix turns out to be a scalar times I, treat that scalar like an ordinary number. Matrix sums then become plain GPs, and (1-t)(t + … + tⁿ) = t - tⁿ⁺¹.

Answer from the NTA final answer key. Solution by PrepWiser.