Let A be a non-singular matrix of order 3 . If det (3 adj (2 adj (( det A) A)))=3⁻¹³ 2⁻¹⁰ and det (3 adj (2 ~A ))=2^ m 3^ n , then |3 ~m +2 n | is equal to

Step-by-step solution

aligned & |3 adj (2 adj (|A| A))|= 3 adj (2|A|^2 adj (A) . & = . |3.2^2 | A |^4 adj ( . adj (A) |=2^6 3^3 | A |¹²|A|^4 . & =2^6 3^3|A|¹⁶=2⁻¹⁰ 3⁻¹³ & |A|¹⁶=2⁻¹⁶ 3⁻¹⁶ |A|=2⁻¹ 3⁻¹ aligned aligned & Now |3 adj (2 ~A )|= |3.2^2 adj ( A ) | & =2^6 3^3| ~A |^2=2^ - m 3^ - n & 2^6 3^3 2⁻² 3⁻²=2^ - m 3^ - n & 2^ - m 3^ - n =2^4 3^1 & m =-4, n =-1 & |3 ~m +2 n |=|-12-2|=14 aligned