2 votes 2 votes A 3 X 3 matrix P has 3 eigen values $-1 , 0.5 , 3$ What will be eigen values of $P^{2} + 2P + I$ Linear Algebra made-easy-test-series engineering-mathematics linear-algebra eigen-value + – jatin khachane 1 asked Dec 2, 2018 edited Mar 4, 2019 by akash.dinkar12 jatin khachane 1 643 views answer comment Share Follow See all 5 Comments See all 5 5 Comments reply sudoankit commented Dec 2, 2018 reply Follow Share The eigenvalues are $0,2.25,16$. Is this correct? 0 votes 0 votes jatin khachane 1 commented Dec 2, 2018 reply Follow Share Yes..how did u solved ? 0 votes 0 votes Lakshman Bhaiya commented Dec 2, 2018 reply Follow Share 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 $\lambda=3$ $\lambda^{2}+2\lambda +1=(3)^{2}+2(3)+1=9+6+1=16$ 2 votes 2 votes sudoankit commented Dec 2, 2018 i edited by sudoankit Dec 2, 2018 reply Follow Share Look here for a good proof (Page 7), https://www.adelaide.edu.au/mathslearning/play/seminars/evalue-magic-tricks-handout.pdf 2 votes 2 votes Lakshman Bhaiya commented Dec 2, 2018 reply Follow Share This is nice pdf thanks!! 0 votes 0 votes Please log in or register to add a comment.