I'm attempting a homework problem and have an idea for proving the result that relies on whether or not matrices with positive entries have only real eigenvalues. Is this true?
For $2\times 2$ matrices this is easy to show. However I'm not sure how to decide if it is true for general $n \times n$ matrices.
I know that the Perron-Frobenius Theorem says the spectral radius of a positive matrix is itself an eigenvalue. If the result I'm asking about is true, can I use this to prove it?