If A is a 3 × 3 matrix and | A | = 2 , then | 3 ? adj ( | 3 A | A 2 ) | is equal to [2023]

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Solution

Q.
If $A$ is a $3 \times 3$ matrix and $| A | = 2$ , then $| 3 adj (| 3 A | A^{2}) |$ is equal to [2023]

1 $3^{11} \cdot 6^{10}$

2 $3^{12} \cdot 6^{11}$

3 $3^{10} \cdot 6^{11}$

4 $3^{12} \cdot 6^{10}$

Ans.
(1)

We have, $| A |$ $= 2 .$

$∴$ $| 3 A |$ $= 3^{3} | A | = 27 \times 2 = 54$

Now, $| 3 adj (| 3 A | A^{2}) |$ $=$ $| 3 adj (54 A^{2}) |$ $=$ $| 3 \times 54^{2} adj (A^{2}) |$

$= {(3 \times 54^{2})}^{3} | adj A |^{2}$

$= 3^{3} \times {(54^{2})}^{3} {(| A |^{2})}^{2} = 3^{3} \cdot 2^{4} \cdot 2^{6} \cdot 27^{6} = 3^{11} 6^{10}$

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