Let A = ( 2, - 2; 1, - 1 ) and B = ( - 1, 2; - 1, 2 ). Then the number of elements in the set (n, m) : n, m ∈ 1, 2, .........., 10 and nAⁿ + mBᵐ = I is ____________.

Solution

Maths sir · 4 steps

Square both matrices first: each one is idempotent, so every power collapses back to the matrix itself.

1. Square A. A² = (4-2, -4+2; 2-1, -2+1) = (2, -2; 1, -1) = A. So Aⁿ = A for every n ≥ 1.
2. Square B. B² = (1-2, -2+4; 1-2, -2+4) = (-1, 2; -1, 2) = B. So Bᵐ = B for every m ≥ 1.

   nA + mB = (2n - m, -2n + 2m; n - m, -n + 2m) = (1, 0; 0, 1)

3. Match entries. The off-diagonal entries give n - m = 0, so n = m. Then the top-left entry gives 2n - n = 1, so n = 1, and the bottom-right entry gives -1 + 2 = 1, which also holds.
4. So. The only pair is (n, m) = (1, 1), and indeed A + B = I. The set has 1 element.

The idea to keep. When you see high powers of a small matrix, compute A² first. If A² = A, every power is just A and the problem becomes plain linear equations.

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