Let A = ( 1, 0, 0; 0, 1, 1; 1, 0, 0 ). Then A²⁰²⁵ - A²⁰²⁰ is equal to :

Solution

Maths sir · 4 steps

Work out A² and A³, spot the pattern in Aⁿ, and the subtraction becomes one number.

1. First few powers. Multiplying gives A² = (1, 0, 0; 1, 1, 1; 1, 0, 0) and A³ = (1, 0, 0; 2, 1, 1; 1, 0, 0). Only the entry in row 2, column 1 changes, and it goes up by 1 each time.

   Aⁿ = (1, 0, 0; n-1, 1, 1; 1, 0, 0), n ≥ 1

2. Subtract two powers. Every entry cancels except row 2, column 1. So Aⁿ - Aᵐ has n - m there and 0 everywhere else. For A²⁰²⁵ - A²⁰²⁰ that entry is 2025 - 2020 = 5.
3. Match the options. A⁶ - A has (6-1) - (1-1) = 5 in that place and zeros elsewhere. A⁵ - A gives only 4. A⁵ and A⁶ alone still have the 1s from Aⁿ, so they are not this matrix.
4. So. A²⁰²⁵ - A²⁰²⁰ = A⁶ - A, which is option (a).

The idea to keep. For a high power of a matrix, compute A² and A³ first. Usually a pattern for Aⁿ appears, and then any question about A²⁰²⁵ is just arithmetic.

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