The characteristic equation of the square matrix is written as, |A−λI|=0 Then, [
1
0
−2
−2
−1
2
3
4
1
]−λ[
1
0
0
0
1
0
0
0
1
]=0 [
1
0
−2
−2
−1
2
3
4
1
]−λ[
λ
0
0
0
λ
0
0
0
λ
]=0 [
1−λ
0
−2
−2
−1−λ
2
3
4
1−λ
]=0 Simplifying the above matrix, we get, (1−λ)[−(1+λ)−8]−2(−8+3(λ+1))=0 (1−λ)[−(1−λ2)−8]−2(−8+3λ+3)=0 (1−λ)(λ2−9)−2(−5+3λ)=0 λ2−9−λ3+9λ−6λ+10=0 Further simplify the above, −λ3+λ2+3λ+1=0 λ3−λ2−3λ−1=0 From the Caley Hamilton’s theorem, A3−A2−3A−I=0 A−1(A3−A2−3A−I)=0 A−1=A2−A−3I