Let A be a square matrix of order 3 such that del(A)=-2 and det(3adj(-6adj(3A)))=2^(m+n). 3^(mn) m>n.

Question

Let A be a square matrix of order 3 such that \(\operatorname{det}(A)=-2\) and \(\operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))=2^{m+n} \cdot 3^{m n}, \mathrm{m}>\mathrm{n}.\) Then \(4 \mathrm{~m}+2 \mathrm{n}\) is equal to ______.

Answer 1

Answer is: 34

\(|\mathrm{A}|=-2\)

\(\operatorname{det}(3 \operatorname{adj}(-6 \operatorname{adj}(3 A)))\)

\(=3^{3} \operatorname{det}(\operatorname{adj}(-\operatorname{adj}(3 \mathrm{~A})))\)

\(=3^{3}(-6)^{6}(\operatorname{det}(3 \mathrm{~A}))^{4}\)

\(=3^{21} \times 2^{10}\)

\(\mathrm{m}+\mathrm{n}=10\)

\(\mathrm{mn}=21\)

\(\mathrm{m}=7 ; \mathrm{n}=3\)