Let A be a 3 × 3 matrix such that
adj A = [ 2, - 1, 1; - 1, 0, 2; 1, - 2, - 1 ] and B = adj(adj A).
If |A| = λ and |(B⁻¹) T | = μ , then the ordered pair,
(|λ|, μ) is equal to :

Solution

Maths sir · 4 steps

Find |adj A| from the given matrix, then let |adj M| = |M|ⁿ⁻¹ do the rest.

1. Determinant of adj A. Expand along the first row: 2(0 + 4) + 1(1 - 2) + 1(2 - 0) = 8 - 1 + 2 = 9. So |adj A| = 9.

   |adj M| = |M|ⁿ⁻¹, n = 3 ⇒ |adj M| = |M|²

2. Find λ. |adj A| = |A|² = 9, so λ² = 9 and |λ| = 3.
3. Find μ. B = adj(adj A), so |B| = |adj A|² = 81. Transpose does not change a determinant, and |B⁻¹| = 1/|B|, so μ = 1/81.
4. So. (|λ|, μ) = (3, 1/81), which is option (c).

The idea to keep. For an n × n matrix, |adj M| = |M|ⁿ⁻¹. Apply it once per adj. Transpose leaves the determinant alone, and an inverse flips it to 1/|M|.

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