Let A be a matrix of order 33 and (A)=2.

Question

Let $A$ be a matrix of order $3	imes3$ and $\det(A)=2$. Then $\det\big(\det(A)\,\operatorname{adj}(5\,\operatorname{adj}(A^3))\big)$ is equal to

Accepted Answer

$\operatorname{adj}(5\operatorname{adj}(A^3))=5^2\operatorname{adj}(\operatorname{adj}(A^3))$, and $|A^3|=|A|^3=8$, so $|\operatorname{adj}(\operatorname{adj}(A^3))|=|A^3|^4=8^4$.

$$\big|\det(A)\operatorname{adj}(5\operatorname{adj}(A^3))\big| = |A|^3\cdot|5\operatorname{adj}(\operatorname{adj}(A^3))| = |A|^3\cdot5^6\cdot|A^3|^4 = 2^3\cdot5^6\cdot2^{12}=2^{15}\cdot5^6.$$

Numerically $2^{15}\cdot5^6 = 32768\times15625=512{,}000{,}000=512\times10^6$.

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