Let A=[2, 1, 0; 1, 2, -1; 0, -1, 2]. If |adj(adj(adj 2 A))|=(16)ⁿ, then n is equal to :

Solution

Maths sir · 4 steps

Find |2A| once, then let the rule |adj B| = |B|² do the work three times.

1. Determinant of A. Expand along the first row: |A| = 2(4 - 1) - 1(2 - 0) + 0 = 6 - 2 = 4.
2. Scale by 2. For a 3 × 3 matrix, |kA| = k³|A|. So |2A| = 8 × 4 = 32 = 2⁵.
3. Three adjoints. For an n × n matrix, |adj B| = |B|ⁿ⁻¹, which is |B|² here. Each adjoint squares the determinant, so three adjoints raise it to the power 2 × 2 × 2 = 8.

   |adj(adj(adj 2A))| = |2A|^((3-1)³) = (2⁵)⁸ = 2⁴⁰

4. So. 2⁴⁰ = (2⁴)¹⁰ = 16¹⁰, so n = 10, which is option (c).

The idea to keep. For an n × n matrix, every adjoint raises the determinant to the power n - 1. Taking the adjoint k times gives |B|^((n-1)ᵏ). Pull the scalar out as kⁿ before you start.

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