Let A be a n × n matrix such that |A|=2. If the determinant of the matrix Adj(2 · Adj(2 A⁻¹)) · is 2⁸⁴, then n is equal to :

Solution

Maths sir · 4 steps

Peel the layers from the inside out, tracking only the power of 2 at each step.

1. Two rules. For an n × n matrix B, |kB| = kⁿ |B| and |adj B| = |B|ⁿ⁻¹. Also |A⁻¹| = 1/|A| = 1/2.
2. Inside out. |2A⁻¹| = 2ⁿ × 1/2 = 2ⁿ⁻¹. Then |adj(2A⁻¹)| = (2ⁿ⁻¹)ⁿ⁻¹ = 2^((n-1)²). Multiplying by 2 gives |2adj(2A⁻¹)| = 2ⁿ × 2^((n-1)²) = 2^(n²-n+1).

   |adj(2adj(2A⁻¹))| = (2^(n²-n+1))ⁿ⁻¹ = 2^((n-1)(n²-n+1))

3. Match powers. Set (n-1)(n²-n+1) = 84. Try values: n = 4 gives 3 × 13 = 39, and n = 5 gives 4 × 21 = 84. The left side keeps increasing for n ≥ 1, so no other n works.
4. So. n = 5.

The idea to keep. A scalar inside a determinant comes out as kⁿ, and each adjoint raises the determinant to the power n-1. Apply these one layer at a time, starting from the innermost bracket.

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