For a 3 × 3 matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a 3 × 3 matrix such that |A|=12and trace (A) = 3. If B = adj(adj(2A)), then the value of |B| + trace (B) equals:

|A|=12,trace(A)=3,B=adj(adj(2A))=|2A|n−2(2A)n=3,B=|2A|(2A)=23⋅|A|(2A)=8A|B|=|8A|=83⋅|A|=28=256trace(B)=8trace(A)=24|B|+trace(B)=280

For a 3 × 3 matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a 3 × 3 matrix such that |A|=12and trace (A) = 3. If B = adj(adj(2A)), then the value of |B| + trace (B) equals:

|A|=12,trace(A)=3,B=adj(adj(2A))=|2A|n−2(2A)n=3,B=|2A|(2A)=23⋅|A|(2A)=8A|B|=|8A|=83⋅|A|=28=256trace(B)=8trace(A)=24|B|+trace(B)=280

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For a 3 × 3 matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a 3 × 3 matrix such that $| A | = \frac{1}{2}$ and trace (A) = 3. If B = adj(adj(2A)), then the value of |B| + trace (B) equals:

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Detailed Solution

$\begin{array}{l} | A | = \frac{1}{2}, trace (A) = 3, B = adj (adj (2 A)) = | 2 A |^{n - 2} (2 A) \\ n = 3, B = | 2 A | (2 A) = 2^{3} \cdot | A | (2 A) = 8 A \\ | B | = | 8 A | = 8^{3} \cdot | A | = 2^{8} = 256 \\ trace (B) = 8 trace (A) = 24 \\ | B | + trace (B) = 280 \end{array}$