Let A=[1, 1/51; 0, 1]. If B=[1, 2; -1, -1] A[-1, -2; 1, 1], then the sum of all the elements of the matrix Σn=1⁵⁰ Bⁿ is equal to

Solution

Maths sir · 4 steps

The outer matrices are inverses, so B = PAP⁻¹ and every power of B is just PAⁿP⁻¹.

1. Spot the inverse. Call P = (1, 2; -1, -1) and Q = (-1, -2; 1, 1). Multiplying gives PQ = I, so Q = P⁻¹ and B = PAP⁻¹.
2. Powers telescope. In Bⁿ = (PAP⁻¹)(PAP⁻¹)… every inner P⁻¹P becomes I, so Bⁿ = PAⁿP⁻¹. Also Aⁿ = (1, n/51; 0, 1), because the top-right entries add up each time you multiply.

   Σₙ₌₁⁵⁰ Aⁿ = (50, (1+2+…+50)/51; 0, 50) = (50, 1275/51; 0, 50) = (50, 25; 0, 50)

3. Conjugate back. The sum is P(Σ Aⁿ)P⁻¹. First P(50, 25; 0, 50) = (50, 125; -50, -75), then multiplying by P⁻¹ gives (75, 25; -25, 25).
4. So. The sum of the elements is 75 + 25 - 25 + 25 = 100, which is option (c).

The idea to keep. Whenever you see PAQ, check whether PQ = I. If it is, then Bⁿ = PAⁿP⁻¹, and a sum of powers of B is P times the same sum of powers of A times P⁻¹.

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