If A is a square matrix of order 3 such that (A)=3 and (adj(-4adj(-3adj(3adj((2A)^-1)))))=…

Question

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

Accepted Answer

$\begin{gathered} |A|=3 \ |adj(-4adj(-3adj(3adj(2A{)}^{-1})))| \ ={|-4adj(-3adj(3adj((2A{)}^{-1})))|}^{2} \end{gathered}$

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

Solutions · 1