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Question Number 98661 by bemath last updated on 15/Jun/20
using cayley − hamilton theorem what is the inverse of matrix A= [((0 1 −1)),((1 2 2)),((0 1 −1)) ]
usingcayley−hamiltontheoremwhatistheinverseofmatrixA=[01−112201−1]
Commented by john santu last updated on 15/Jun/20
we first compute the characteristic equation ∣A−λI∣=0 [((−λ 1 −1 )),(( 1 2−λ 2)),(( 0 1 −1−λ)) ]= 0 = −λ{(2−λ)(−1−λ)−2}−(−1−λ)−1 = λ{λ^2 −λ−4}+1+λ−1 = λ^3 −λ^2 −3λ the cayley−hamilton theorem states that a matrix satisfies its own characteristic equation ⇒A^3 −A^2 −3A = 0 A(A^2 −A−3I) = 0 ⇔I = (1/3)A(A−I) ⇔A^(−1) = (1/3)(A−I) ⇔A^(−1) = (1/3) [((−1 1 −1)),(( 1 1 2)),(( 0 1 −2)) ]
wefirstcomputethecharacteristicequation∣A−λI∣=0[−λ1−112−λ201−1−λ]=0=−λ{(2−λ)(−1−λ)−2}−(−1−λ)−1=λ{λ2−λ−4}+1+λ−1=λ3−λ2−3λthecayley−hamiltontheoremstatesthatamatrixsatisfiesitsowncharacteristicequation⇒A3−A2−3A=0A(A2−A−3I)=0⇔I=13A(A−I)⇔A−1=13(A−I)⇔A−1=13[−11−111201−2]

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