Let A be a 3 3 matrix such that aligned & | adj ( adj ( adj A ))|=81 . If & S = n Z :(| adj ( adj A)|)^ (n-1)^2 2 =|A|^ (3 n^2-5 n-4 ) aligned , then _ n S |A^ (n^2+n ) | is equal to

Step-by-step solution

aligned & | adj ( adj )( adjA )|=81 & | adjA |^4=81 & | adjA |=3 & | ~A |^2=3 & | ~A |= 3 & (| ~A |^4 )^ ( n -1)^2 2 =| A |^ n ^2-5 n -4 & 2( n -1)^2=3 n ^2-5 n -4 & 2 n ^2-4 n +2=3 n ^2-5 n -4 & n ^2- n -6=0 & ( n -3)( n +2)=0 & n =3,-2 & _ n s | A ^ n ^2+ n | & = | A ^2 |+ | A ¹² | & =3+36=3+729=732 aligned