9 votes 9 votes A 3X3 matrix P is such that, P^3 = P. Then the eigenvalues of P are: 1) 1, 1, -1 2) 1, 0.5 + j0.866, 0.5 - j0.866 3) 1, -0.5 + j0.866, -0.5 - j0.866 4) 0, 1, -1 Linear Algebra gate2016-ee-2 linear-algebra eigen-value + – Shubhanshu asked May 19, 2017 Shubhanshu 5.3k views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
Best answer 11 votes 11 votes Suppose $x$ is an eigenvector of $P$, and let $\lambda$ be corresponding eigenvalue, then $$Px=\lambda x$$ Also $$P^3x=\lambda^3 x$$. We are given $P^3=P$, which means $$P^3x=Px \Rightarrow \lambda^3 x = \lambda x \Rightarrow (\lambda^3-\lambda)x=0$$ Since $x$ is non-zero, $\lambda^3-\lambda = 0$. Solving it, we get $\lambda={0,1,-1}$ Happy Mittal answered May 24, 2017 selected Aug 16, 2017 by Shubhanshu Happy Mittal comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes option 4?? joshi_nitish answered May 19, 2017 joshi_nitish comment Share Follow See all 4 Comments See all 4 4 Comments reply Shubhanshu commented May 19, 2017 reply Follow Share Yes but how? 0 votes 0 votes joshi_nitish commented May 19, 2017 reply Follow Share every square matrix saisfies its own characterstic equation...therefore, λ^3-λ=0 λ=0,1,-1.. but since i am deriving characterstic equation back from given matrix condition, i am not sure about it. 2 votes 2 votes Shubhanshu commented May 20, 2017 reply Follow Share then why not option 1 ??? 0 votes 0 votes joshi_nitish commented May 20, 2017 reply Follow Share @shubhanshu because roots of λ^3-λ=0 will be 0,1,-1... 0 votes 0 votes Please log in or register to add a comment.