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please help me with this determinant problem from mains

please help me with this determinant problem from mains
 

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Grade:12

1 Answers

Aditya Gupta
2081 Points
5 years ago
we know from the standard formula:
adj(Ak) = (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

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