Let A=[2, -1; 1, 1]. If the sum of the diagonal elements of A¹³ is 3ⁿ, then n is equal to ________.

Solution

Maths sir · 4 steps

Do not multiply 13 times. Find a small power of A that becomes a multiple of I, then the rest is quick.

1. Square A. A² = (2, -1; 1, 1)(2, -1; 1, 1) = (3, -3; 3, 0).
2. Cube A. A³ = A · A² = (3, -6; 6, -3) = 3B, where B = (1, -2; 2, -1). Squaring B gives B² = (-3, 0; 0, -3) = -3I.

   A⁶ = (3B)² = 9(-3I) = -27I ⇒ A¹² = 729I = 3⁶ I

3. Reach the 13th power. A¹³ = A¹² · A = 3⁶ A. The diagonal elements of A add up to 2 + 1 = 3.
4. So. The sum of the diagonal elements of A¹³ is 3⁶ × 3 = 3⁷, so n = 7.

The common slip: 6. Stopping at A¹³ = 3⁶ A and reading the exponent off gives 6. The diagonal elements of A itself add up to 3, which adds one more power of 3.

The idea to keep. For a high power of a 2 × 2 matrix, keep multiplying until some power turns into a multiple of I. After that, every larger power is that scalar times a small power you already know.

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