Let A be a matrix of order 3 imes 3 and |A|=5. If |2 \operatorname{adj}(3 A \operatorname{adj}(2 A))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma, \alpha, \beta, \gamma \in N, then \alpha+\beta+\gamma is equal to

Question

Let A be a matrix of order 3 	imes 3 and |A|=5. If |2 \operatorname{adj}(3 A \operatorname{adj}(2 A))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma, \alpha, \beta, \gamma \in N, then \alpha+\beta+\gamma is equal to

Accepted Answer

Answer: B. 27 To find the expression |2 \operatorname{adj}(3 A \operatorname{adj}(2 A))|, we break it down as follows: Recognize that: |2 \operatorname{adj}(3 A \operatorname{adj}(2 A))| = 2^3 |3A (\operatorname{adj}(2A))|^2 Apply properties of determinants: = 2^3 (3^3)^2 |A|^2 \left|\operatorname{adj}(2A)ight|^2 Further simplify using |\operatorname{adj}(B)| = |B|^{n-1} for a 3 	imes 3 matrix: = 2^3 \cdot 3^6 \cdot 5^2 \cdot (|2A|^2)^2 Simplify |2A|: = 2^3 \cdot 3^6 \cdot 5^2 \cdot (2^3)^4 \cdot |A|^4 Continue to simplify: = 2^3 \cdot 3^6 \cdot 5^2 \cdot (2^3)^4 \cdot 5^4 Expand and combine powers: = 2^{15} \cdot 3^6 \cdot 5^6 Therefore, \alpha = 15, \beta = 6, and \gamma = 6. So, \alpha + \beta + \gamma = 15 + 6 + 6 = 27.

Let $A$ be a matrix of order $3 \times 3$ and $|A|=5$. If $|2 \operatorname{adj}(3 A \operatorname{adj}(2 A))|=2^\alpha \cdot 3^\beta \cdot 5^\gamma, \alpha, \beta, \gamma \in N$, then $\alpha+\beta+\gamma$ is equal to

Answer

Explanation

More JEE Math questions