Let A be a non-singular matrix of order 3. If det(3adj(2adj((detA)A))) = 3<sup>-13</sup> ⋅ 2<sup>-10</sup> and det(3adj(2A)) = 2<sup>m</sup> ⋅ 3<sup>n</sup>, then |3m + 2n| is equal to ____.
Step-by-Step Solution
Key Concept: Use properties of determinants: det(kA) = k^n det(A), det(adj(A)) = (det(A))^(n-1), and det(adj(kA)) = (k^(n-1) det(A))^(n-1) for a matrix of order n.
Let det(A) = Δ. Order n = 3. <br>1) det(3adj(2adj(ΔA))) = 3<sup>3</sup> det(adj(2adj(ΔA))) = 27 (det(2adj(ΔA)))<sup>2</sup> <br>det(2adj(ΔA)) = 2<sup>3</sup> det(adj(ΔA)) = 8 (Δ<sup>3-1</sup>)<sup>3-1</sup> = 8 Δ<sup>4</sup> <br>So, 27 (8 Δ<sup>4</sup>)<sup>2</sup> = 27 ⋅ 64 ⋅ Δ<sup>8</sup> = 3<sup>3</sup> ⋅ 2<sup>6</sup> ⋅ Δ<sup>8</sup> = 3<sup>-13</sup> ⋅ 2<sup>-10</sup> <br>Δ<sup>8</sup> = 3<sup>-16</sup> ⋅ 2<sup>-16</sup> = (3<sup>-2</sup> ⋅ 2<sup>-2</sup>)<sup>8</sup>. Thus Δ = ± 1/6. <br>2) det(3adj(2A)) = 3<sup>3</sup> det(adj(2A)) = 27 (det(2A))<sup>2</sup> <br>det(2A) = 2<sup>3</sup> Δ = 8Δ. <br>det(3adj(2A)) = 27 (8Δ)<sup>2</sup> = 27 ⋅ 64 ⋅ Δ<sup>2</sup> = 3<sup>3</sup> ⋅ 2<sup>6</sup> ⋅ (1/36) = 3<sup>3</sup> ⋅ 2<sup>6</sup> ⋅ 3<sup>-2</sup> ⋅ 2<sup>-2</sup> = 3<sup>1</sup> ⋅ 2<sup>4</sup>. <br>So m=4, n=1. |3m + 2n| = |12 + 2| = 14.
Correct Answer: 14