Let A be a square matrix of order 3 such that det(A)=-2 and det(3 adj(-6 adj(3 A)))=2ᵐ⁺ⁿ · 3ᵐⁿ, m>n. Then 4 m+2 n is equal to __________.

Solution

Maths sir · 4 steps

Peel the nest from the inside out with two rules: scaling gives k³, adjoint gives a square.

1. Two rules. For an order-3 matrix, det(kM) = k³ det M and det(adj M) = (det M)². Apply them one layer at a time, starting from 3A.
2. Inner layers. det(3A) = 27 × (-2) = -54, so det(adj(3A)) = (-54)² = 2² · 3⁶. Then det(-6adj(3A)) = (-6)³ · 2² · 3⁶ = -2⁵ · 3⁹.
3. Outer layers. Squaring for the next adjoint removes the minus sign: (2⁵ · 3⁹)² = 2¹⁰ · 3¹⁸. Multiplying by 3 outside the adjoint brings in 3³, giving 2¹⁰ · 3²¹.

   2ᵐ⁺ⁿ· 3ᵐⁿ = 2¹⁰· 3²¹ ⇒ m+n=10, mn=21

4. So. The two numbers are 7 and 3. Since m > n, m = 7 and n = 3, so 4m + 2n = 28 + 6 = 34.

The common slip: 26. Ignoring the condition m > n and taking m = 3, n = 7 gives 12 + 14 = 26, not 34.

The idea to keep. For an order-n matrix, a scalar outside the determinant comes out as kⁿ, and each adjoint raises the determinant to the power n-1. Work through nested expressions one layer at a time.

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