please help me with this determinant problem from mains

we know from the standard formula:

adj(A^k) = (adj(A))^k

so that adj((I – A)⁶) = (adj(I – A))⁶

so det[adj((I – A)⁶)]= det[(adj(I – A))⁶]= [det(adj(I – A))]⁶

but we also know from standard formulas that

det(adj(A))= (det(A))^n-1, n being the order of matrix A.

so det(adj(I – A))= (det(I – A))^2-1= det(I – A)

or [det(adj(I – A))]^2= [det(I – A)]^2= det((I – A)^2)= det(I+A^2 – 2A)= det( – 3A)= (– 3)^2*det(A)= 3^2 (because det(A)=1 as can be obtained by multiplying A^2+A+I=0 by A – I on both sides to obtain A^3=I then taking det on both sides).

or [det(adj(I – A))]^6= 3^6