If A is a square matrix of order 3 such that [det(A)=3] and [det(adj(-4 adj(-3 adj(3 adj((…

Question

If $A$ is a square matrix of order 3 such that $$\operatorname{det}(A)=3$$ and $$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$$, then $$\mathrm{m}+2 \mathrm{n}$$ is equal to :

Accepted Answer

$$\begin{aligned}
& |A|=3 \
& \left|\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}(2 A)^{-1}\right)\right)\right)\right| \
& =\left|-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 A)^{-1}\right)\right)\right)\right|^2
\end{aligned}$$

$$\begin{aligned}
& =4^6\left|-3 \operatorname{adj}\left(\operatorname{aadj}\left((2 A)^{-1}\right)\right)\right|^4 \
& =4^6 \cdot 3^{12}\left|\operatorname{aadj}\left((2 A)^{-1}\right)\right|^8 \
& =4^6 \cdot 3^{12} \cdot 3^{24}\left|(2 A)^{-1}\right|^{16} \
& =4^6 \cdot 3^{36} 2^{-48}\left|A^{-1}\right|^{16} \
& =\frac{2^{-36} 3^{36}}{3^{16}}=2^{-36} 3^{20} \
& m=-36 \
& n=20 \
& m+2 n=4
\end{aligned}$$

Solutions · 1