The characteristic equation of a 3×3 matrix P is defined as a(λ)=|λI−P|=λ3+λ2+2λ+1=0
If I denotes identify matrix, then the inverse of matrix P will be
A
(P2+P+2I)
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B
(P2+P+1)
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C
−(P2+P+I)
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D
−(P2+P+2I)
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Solution
The correct option is D−(P2+P+2I) According to Cayley-Hamilton theorem, every square matrix satisfies its own characteristic equation ⇒P3+P2+2P+1=O
Multiplying by matrix [P]−1 ⇒P2+P+2I+P−1=0 ⇒P−1=−(P2+P+2I)