Mathematics - Matrices Question with Solution | TestHub

$\begin{array}{rl}& \text{ As }A\mathrm{adj}A=|A|l,\det (\lambda A)={\lambda }^n\det A\\ & \det (3\mathrm{adj}(-6\mathrm{adj}(3A)))=3^3\det (\mathrm{adj}(-6\mathrm{adj}(3A)))\\ & =3^3(-6\mathrm{adj}(3A){)}^2\\ & =3^3(-6{)}^6|3A{|}^4\\ & =3^9{2}^6\cdot 3^{12}\cdot (-2{)}^4\\ & =3^{21}\cdot 2^{10}\end{array}$ Now comparing with given condition $\begin{array}{rl}& 2^{m+n}{3}^{mn}=2^{10}\cdot 3^{21}\\ & m+n=10,mn=21\\ & \Rightarrow m=7,n=3(m>n)\\ & \therefore 4m+2n=28+6=34\end{array}$