Let A be a non-singular matrix of order 3. If det(3 adj(2 adj((det A) A)))=3⁻¹³ · 2⁻¹⁰ and det(3adj(2 A))=2ᵐ · 3ⁿ, then |3 m+2 n| is equal to _________.

Solution

Maths sir · 4 steps

Peel the expression from the inside out with two rules: k³ for scalars and a square for adj.

1. Two rules. For a matrix X of order 3, det(kX) = k³ det X and det(adj X) = (det X)². Let d = det A.
2. Inside out. First, det(dA) = d³ · d = d⁴. Its adjoint has determinant (d⁴)² = d⁸. Multiplying by 2 gives 2³ d⁸. The next adjoint gives (2³ d⁸)² = 2⁶ d¹⁶. Multiplying by 3 gives 3³ · 2⁶ d¹⁶.

   3³ · 2⁶ d¹⁶ = 3⁻¹³ · 2⁻¹⁰ ⇒ d¹⁶ = 3⁻¹⁶ · 2⁻¹⁶ ⇒ d² = 1/36

3. Second determinant. det(3adj(2A)) = 3³ (det 2A)² = 27 (8d)² = 3³ · 2⁶ · d² = 3³ · 2⁶ · 2⁻² · 3⁻² = 2⁴ · 3.
4. So. m = 4 and n = 1, so |3m + 2n| = |12 + 2| = 14.

The idea to keep. In order n, a scalar comes out of the determinant as kⁿ and adj raises the determinant to the power n-1. For nested expressions, always work from the innermost bracket outward.

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