Choose correct answers on eigen value:

Question

A =

[0 0 L 0 1
 1 0 L 0 0
 0 1 L 0 0
 M M N N M
 0 0 L 1 0]

a) eigenvalue are purely real b) 0 is the only eigenvalue c) eigenvalues are n-th roots of unity Exp(2πi/n) for i = 0,1,...,n-1 d) none of these

What can be the easiest way to get answers in these type of questions?

Answer 1

I think that the best option is really to pick nice values for $N$, $M$, and $L$ and try to compute. This matrix doesn't appear to have enough structure to answer this type of question quickly (although I might be missing something).

Sketch:

1.) Compute eigenvalues for $L=1$, $M=0$, and $N=1$. This gets rid of two possibilities (note that the rows sum to $2$).

2.) Compute the eigenvalues for $L=0$, $M=0$, and $N=1$. This gets rid of one more case (note that this choice eliminates most of the terms in the characteristic equation and Descartes' rule of signs can be applied to the polynomial to count the real roots).