Let S = { n ∈ N| ( 0, i; 1, 0 )ⁿ( a, b; c, d ) = ( a, b; c, d )∀ a,b,c,d ∈ R }, where i = √(- 1). Then the number of 2-digit numbers in the set S is _____________.

Solution

Maths sir · 4 steps

Square the matrix first: A² turns out to be iI, and then the powers are easy to follow.

1. What S asks. Call the first matrix A. If Aⁿ M = M for every real matrix M, then taking M = I gives Aⁿ = I. So S is the set of n with Aⁿ = I.

   A² = (0, i; 1, 0)(0, i; 1, 0) = (i, 0; 0, i) = iI

2. Find the cycle. For an even power, A²ᵏ = iᵏ I, which equals I only when iᵏ = 1, that is when k is a multiple of 4. For an odd power, A²ᵏ⁺¹ = iᵏ A, which has zeros on the diagonal and can never be I. So Aⁿ = I exactly when n is a multiple of 8.
3. Count. The two-digit multiples of 8 are 16, 24, …, 96. That gives (96 - 16)/8 + 1 = 11 numbers.
4. So. The number of 2-digit numbers in S is 11.

The common slip: 22. Stopping at A⁴ = I gives multiples of 4 and a count of 22. In fact A⁴ = i² I = -I, so the cycle length is 8, not 4.

The idea to keep. When a matrix power must leave every matrix unchanged, it must equal I. Compute A² first: if it is a scalar times I, the full cycle length follows directly.

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