If A = [(2 2+p 2+p+q), (4 +62p 8+3p+2q), (6 12+3p 20+6p+3q)], then the value of det(adj(adj(3A))) = 2^m∙3^n, then m + n is equal to

Question

If \(A=\left[\begin{array}{ccc}2 & 2+p & 2+p+q \\ 4 & 6+2 p & 8+3 p+2 q \\ 6 & 12+3 p & 20+6 p+3 q\end{array}\right],\) then the value of \(\operatorname{det}(\operatorname{adj}(\operatorname{adj}(3 A)))=2^{m} \cdot 3^{n},\) then m + n is equal to

(1) 20

(2) 24

(3) 36

(4) 18  

Answer 1

Correct option is: (2) 24  

\(\left|\begin{array}{ccc}2 & 2+p & 2+p+q \\ 4 & 6+2 p & 8+3 p+2 q \\ 6 & 12+3 p & 20+6 p+3 q\end{array}\right|\)  

\(=8(1(6)-1(8)+1(3)) \)

= 8

\( |\operatorname{adj}(\operatorname{adj}(3 A))|=(|3 A|)^{2^{2}}=|3 A|^{4} \)

\(=\left(3^{3}|A|\right)^{4}=3^{12} \cdot|A|^{4} \)

\(=3^{12} \cdot\left(2^{3}\right)^{4}\)  

\(=3^{12} \cdot 2^{12}\)