Algebraic multiplicity and similarity between rows

Question

I know that if two rows in a square matrix are identical, one eigenvalue will have multiplicity of at least 2. I was wondering if two rows are very similar in their elements but not identical it would imply that the matrix will have two similar but not identical eigenvalues. Is that correct?

Thanks!

(Corrected): in the matrix all elements are positive and nonzero

Answer 1

Your statement is not true. For example, $$\pmatrix{0 & 1\cr 0 & 1\cr}$$ has two identical rows, and its eigenvalues $0$ and $1$ both have multiplicity $1$.

All you can conclude from the matrix having two identical rows is that the matrix is singular, i.e. $0$ is an eigenvalue.