Let a R and A be a matrix of order 3 3 such that det (A)=-4 and A+I= [ array lll 1 & a & 1 2 & 1 & 0 a & 1 & 2 array ] , where I is the identity matrix of order 3 3 . If det ((a+1) adj ((a-1) A)) is 2^m 3^n, m, n 0,1,2, .20 , then m + n is equal to :

Step-by-step solution

A = [ array lll 1 & a & 1 2 & 1 & 0 a & 1 & 2 array ]- I = [ array lll 0 & a & 1 2 & 0 & 0 a & 1 & 1 array ] aligned & | A |=-4 2-2 a =-4 a =3 & |( a +1) adj ( a -1) A |=|4 adj 3 ~A | & =4^3| adj 3 ~A | & =4^3 |3 ~A |³⁻¹=64|3 ~A |^2 & =64 (3^3 )^2| ~A |^2 & =2^6 3^6 16 & 2^ m 3^ n =2¹⁰ 3^6 & ~m =10, n =6 & ~m + n =16 aligned