Let A be a matrix of order 3 times 3 and det( | vedclass

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(A) Given $A$ is a $3 	imes 3$ matrix,so $n = 3$. We have $\det(A) = 2$.We need to find $\det(\det(A) \cdot \operatorname{adj}(5 \operatorname{adj}(A

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$512 	imes 10^6$

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(A) Given $A$ is a $3 	imes 3$ matrix,so $n = 3$. We have $\det(A) = 2$.We need to find $\det(\det(A) \cdot \operatorname{adj}(5 \operatorname{adj}(A^3)))$.Since $\det(A) = 2$,the expression becomes $\det(2 \cdot \operatorname{adj}(5 \operatorname{adj}(A^3)))$.Using the property $\det(kA) = k^n \det(A)$,we get $2^3 \det(\operatorname{adj}(5 \operatorname{adj}(A^3)))$.Using $\det(\operatorname{adj}(M)) = (\det(M))^{n-1}$,where $n=3$,we have $8 \cdot (\det(5 \operatorname{adj}(A^3)))^2$.Using $\det(kM) = k^n \det(M)$,we have $8 \cdot (5^3 \det(\operatorname{adj}(A^3)))^2 = 8 \cdot 5^6 \cdot (\det(\operatorname{adj}(A^3)))^2$.Using $\det(\operatorname{adj}(M)) = (\det(M))^{n-1}$,we have $8 \cdot 5^6 \cdot ((\det(A^3))^2)^2 = 8 \cdot 5^6 \cdot (\det(A)^3)^4$.Substituting $\det(A) = 2$,we get $2^3 \cdot 5^6 \cdot (2^3)^4 = 2^3 \cdot 5^6 \cdot 2^{12} = 2^{15} \cdot 5^6$.$2^{15} \cdot 5^6 = 2^9 \cdot 2^6 \cdot 5^6 = 512 \cdot 10^6$.

Let $A$ be a matrix of order $3 \times 3$ and $\det(A) = 2$. Then $\det(\det(A) \cdot \operatorname{adj}(5 \operatorname{adj}(A^3)))$ is equal to:

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