Consider a positive matrix sequence $\{M_1, M_2, \cdots, M_k\}$ where $M_{1} = \begin{bmatrix}x_{1, 1}\end{bmatrix}$, $M_2 = \begin{bmatrix} x_{1, 1} & x_{1, 2} \\ x_{2, 1} & x_{2, 2}\end{bmatrix}$, $\cdots$, $M_k = \begin{bmatrix} x_{1, 1} & x_{1, 2} & \cdots & x_{1, k} \\x_{2, 1} & x_{2, 2} & \cdots & x_{2, k} \\ \vdots & \vdots & \ddots & \vdots \\ x_{k, 1} & x_{k, 2} & \cdots & x_{k, k} \end{bmatrix} = \begin{bmatrix}p & p & p& \cdots & p & p \\ p^2 & p(1+p) & p(1+p) & \cdots & p(1+p) & p(1+p) \\ p^2(1+p) & p^2(2+p) & p(1+p)^2 & \cdots & p(1+p)^2 &p(1+p)^2\\ p^2(1+p)^2 & p^2(1+p)(2+p) & p(1+p)^3 - p & \cdots & p(1+p)^3& p(1+p)^3 \\ p^2(1+p)^3 & p^2(1+p)^2(2+p) & p(1+p)^4 - p(1+p) & \cdots & p(1+p)^4& p(1+p)^4 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ p^2(1+p)^{k-2} & p^2(1+p)^{k-3}(2+p) & p(1+p)^{k-1} - p(1+p)^{k-4} & \cdots &p(1+p)^{k-1} - p & p(1+p)^{k-1}\end{bmatrix}$, where $ p> 0$ and thus $x_{i, j} > 0, \forall i, j=1, 2, \cdots, k$. we want to show that the spectral radius is strictly increasing, i.e. $\rho(M_{d}) < \rho(M_{d+1}), \forall d=1, 2, \cdots, k-1$. Also we are interested in the changes of smallest eigenvalue, can we show that they are decreasing?
Now I know $x_{i, 1} < x_{i, 2} < x_{i, k}, \forall i=1, 2, \cdots, k$, so $\sum\limits_{j=1}^{d}x_{j, 1} \leq \rho(M_d) \leq \sum\limits_{j=1}^{d}x_{j, d}, \forall d=1, 2, \cdots, k $, but $\sum\limits_{j=1}^{d}x_{j, d} > \sum\limits_{j=1}^{d+1}x_{j, 1}$ sometimes, so I cannot use this relation to show $\rho(M_d) < \rho(M_{d+1})$.
Is there any other ways to show the strictly increasing spectral radius?