verify cayley hamilton theorem and find its inverse of the following matrix [(1,4,1),(3,2,2),(7,3,1)]
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To verify C- H theorem characteristic eqn |A- λI| =0
i.e. ∣∣∣∣1−λ−1023−λ−2−201−λ∣∣∣∣=0|1−λ2−2−13−λ00−21−λ|=0
Expanding we get a cubic equation in λ as λ3−5λ2+9λ−1=0λ3−5λ2+9λ−1=0
To verify C-H theorem we have to show that A3−5A2+9A−I=0A3−5A2+9A−I=0 ………………..(1)
Taking L.H.S. of eqn (1)
A3−5A2+9A–I=⎡⎣⎢−13−1110429−22−210−3⎤⎦⎥−5⎡⎣⎢−1−42127−8−421⎤⎦⎥+9⎡⎣⎢1−1023−2−201⎤⎦⎥−
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Answer:
The answer is
A-1 = 1/8 [2/-6. -1/1]
is the correct answer
Thank you
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