Let A be a matrix of order 3 × 3 and |A|=5. If |2 adj(3 A adj(2 A))|=2^α · 3^β · 5^γ, α, β, γ ∈ N, then α+β+γ is equal to

Solution

Maths sir · 4 steps

Use A adj A = |A| I to collapse the inner part to a scalar matrix, and the rest is quick.

1. Pull out the 2. For a 3 × 3 matrix, adj(kA) = k²adjA. So adj(2A) = 4adjA.
2. Collapse the inside. Let B = 3Aadj(2A) = 12 AadjA. Since AadjA = |A| I = 5I, we get B = 60I.
3. Adjoint of B. For B = 60I, adjB = 60²I = 3600I. So 2adjB = 7200I.

   |7200I| = 7200³ = (2⁵· 3²· 5²)³ = 2¹⁵· 3⁶· 5⁶

4. So. α = 15, β = 6, γ = 6, and α + β + γ = 27, which is option (b).

The idea to keep. Whenever A and adjA are multiplied together, replace the product with |A|I at once. Remember that |kM| = kⁿ|M|, so a scalar comes out cubed for a 3 × 3 matrix.

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