Tridiagonal matrix power with negative diagonal

Question

I have a tridiagonal matrix with negative diagonal and and non-negative off-diagonal elements where every row sums to zero (it is the generator to an M/M/n/n queue). My experiments indicate that $|A|^p $ = $|A^p|$, where $|\cdot|$ is the absolute value of every element, but I am not able to prove this.

Answer 1

Let $D$ denote the diagonal matrix with entries $(-1)^i$ on the main diagonal, i.e. $$d_{ij}=\begin{cases}(-1)^i & j=i\\ 0 & j\neq i \end{cases} $$ Observe that the matrix $DAD$ has nonpositive entries and $|DAD|=|A|.$ Therefore $|A|=-DAD.$ We have $D^2=I.$ Hence $(DAD)^k=DA^kD.$ Therefore $|(DAD)^k|=|DA^kD|=|A^k|.$ Thus $$|A|^k=(-DAD)^k=(-1)^kDA^kD=|DA^kD|=|A^k|$$

Remark The assumption that the rows sum up to $0$ is not necessary for the conclusion.