For a matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a matrix such that and trace (A) = 3. If B = adj (adj (2A)), then the value of |B| + trace (B) equals : [2025]

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Solution

Q.
For a $3 \times 3$ matrix M, let trace (M) denote the sum of all the diagonal elements of M. Let A be a $3 \times 3$ matrix such that $| A |$ $= \frac{1}{2}$ and trace (A) = 3. If B = adj (adj (2A)), then the value of $| B |$ + trace (B) equals : [2025]

1 56

2 280

3 132

4 174

Ans.
(2)

Given, $| A |$ $= \frac{1}{2}$ , trace (A) = 3

B = adj (adj (2A)) [ $∵ adj (adj A) = | A |^{n - 2} A$ ]

$= | 2 A |^{3 - 2} (2 A) = 2^{3} | A | (2 A) = 8 A$

$\Rightarrow trace (B) = 8 trace (A) = 8 \times 3 = 24$

Hence, $| B |$ + trace (B)

$= 8^{3}$ $| A |$ $+ 24 = \frac{8^{3}}{2} + 24 = 280$ .

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