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A 3 X 3 matrix P has 3 eigen values $-1 , 0.5 , 3$
What will be eigen values of $P^{2} + 2P + I$

According to Cayley Hamilton theorem: Every Square matrix satisfies its own characteristic equation.

So$, P=\lambda I$

Given that $\lambda_{1}=0,\lambda_{2}=0.5,\lambda_{3}=3$

$P^{2}+2P+I=(\lambda I)^{2}+2(\lambda I)+I=\lambda ^{2}+2\lambda +1$

Put $\lambda=0$

$\lambda^{2}+2\lambda +1 = 0+0+1=1$

Put $\lambda=0.5$

$\lambda^{2}+2\lambda_{3} +1=(0.5)^{2}+2(0.5)+1=0.25+1+1=2.25$

Put $\lambda=3$

$\lambda^{2}+2\lambda +1=(3)^{2}+2(3)+1=9+6+1=16$

edited
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