Let A be a matrix of order and |A| = 5. If , then is equal to [2025]

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Solution

Q.
Let A be a matrix of order $3 \times 3$ and |A| = 5. If $| 2 a d j (3 A a d j (2 A)) | = 2^{α} \cdot 3^{β} \cdot 5^{γ}, α, β, γ \in N$ , then $α + β + γ$ is equal to [2025]

1 28

2 25

3 27

4 26

Ans.
(3)

We have, |2 adj(3A adj(2A))|

$= 2^{3} \cdot$ $| 3 A adj (2 A) |^{2}$ [ $∵ | adj A | = | A |^{n - 1}$ , when n is order of matrix A]

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

$= 2^{3} \cdot 3^{6} \cdot | A |^{2} \cdot {(| 2 A |^{2})}^{2}$

$= 2^{3} \cdot 3^{6} \cdot | A |^{2} [{(2)}^{6} \cdot | A |^{2}]^{2}$

$= 2^{3} \cdot 3^{6} \cdot | A |^{2} \cdot 2^{12} \cdot | A |^{4} = 2^{15} \cdot 3^{6} \cdot | A |^{6}$

$= 2^{15} \cdot 3^{6} \cdot 5^{6} = 2^{α} \cdot 3^{β} \cdot 5^{γ}$ [ $∵$ $| A |$ = 5]

By comparing, we get

$α = 15, β = 6, γ = 6$

$∴ α + β + γ = 27$ .

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