Let $T$ be an $n \times n$ row-stochastic matrix which is primitive (i.e. there is a positive integer $k$ such that all entries of $T^k$ are strictly positive). Let $T_i$ denote the matrix obtained by eliminating the $i$th row and $i$th column from $T$. How do I prove that $$\det(I - T_i) > 0 \quad \text{for all } i \; ?$$
($I$ denotes the $(n-1) \times (n-1)$ identity matrix.)
Here are some facts that may be relevant:
1 is an eigenvalue of $T$.
If $\lambda$ is an eigenvalue of $T$ then $|\lambda| \leq 1$.
If $\lambda_1,\lambda_2,\ldots,\lambda_n$ are the eigenvalues of $T_i$, then $1-\lambda_1,1-\lambda_2,\ldots,1-\lambda_n$ are the eigenvalues of $I-T_i$.
A sufficient condition for $\det(I-T_i)>0$ is to have all eigenvalues of $I-T_i$ be strictly positive. So in light of the previous fact, it sufficies to show that the eigenvalues of $T_i$ are all strictly less than 1. But I don't know how to rigorously establish this.
for your problem, which is an offshoot of this:
Question about a proof of the Perron Frobenius Theorem
it suffices to consider the case where $T$ is strictly positive (componentwise) and let $P:=T$. If this is unsatisfactory consider using $P:=\frac{1}{n}\big(T+T^2+.... T^n\big)$ which must be strictly positive componentwise, and observe that $P$ and $T$ have the same algebraic multiplicity of $\lambda_1 =1$.
Now $P\mathbf 1_n = \mathbf 1_n$ and
$P_i\mathbf 1_{n-1} \lt \mathbf 1_{n-1}$ (component-wise)
$\longrightarrow \text{spectral radius}(P_i) =\sigma(P_i)\lt 1$ by Gerschgorin Discs.
In general $\det(Ix - B)$ is a monic real polynomial in x and thus its image is $\gt 0$ for large enough $x$. Let $B:= P_i$. We know $\det(I - P_i)\neq 0$ since $\sigma(P_i)\lt 1$ and if $\det(I - P_i)\lt 0$ then by intermediate value theorem there is some $x'\gt 1$ such that $\det(Ix' - P_i)= 0$ but this contradicts $\sigma(P_i)\lt 1$.
We conclude $\det(I - P_i)\gt 0$.
edit: addendum
to explain in a very granular way as to why the algebraic multiplicity of $\lambda_1=1$ is the same for $P=\frac{1}{n}\big(T+T^2+.... T^n\big)$ and $T$
0.) for upper triangular matrix $R$, $\det\big(xI - R\big)=\prod_{j=1}^n (x-r_{j,j})$ so the algebraic multiplicity of some eigenvalue comes down to counting how many times it shows up on the diagonal
1.) over $\mathbb C$ every matrix is similar to an upper triangular matrix (e.g. using Schur Triangularization for a light weight approach), and similarity transforms do not affect the char poly/eigenvalues
2.) use a similarity transform to triangularize $T$
$R= S^{-1}TS \longrightarrow U= S^{-1}PS = \frac{1}{n}\big(R+R^2+.... + R^n\big)$
where $U$ is necessarily upper triangular
3.) $\text{eigenvalue j of R is 1} \longrightarrow \text{eigenvalue j of U is 1}$
$u_{j,j}= \frac{1}{n}\big(r_{j,j}+r_{j,j}^2+....+ r_{j,j}^n\big) = \frac{1}{n}\big(1+1+.... + 1\big) =1$
4.) $\text{eigenvalue j of R is not 1} \longrightarrow \text{eigenvalue j of U is not 1}$
to make this extra clear, split into 2 cases, and in each case apply triangle inequality
i.) $\vert r_{j,j}\vert \lt 1$
$\vert u_{j,j}\vert $
$= \frac{1}{n}\Big\vert r_{j,j}+r_{j,j}^2+....+ r_{j,j}^n\Big\vert $
$\leq \frac{1}{n}\Big(\big\vert r_{j,j}\vert +\vert r_{j,j}\vert ^2+....+ \vert r_{j,j}\vert^n\Big)$
$\leq \frac{1}{n}\Big(\big\vert r_{j,j}\vert +\vert r_{j,j}\vert +....+ \vert r_{j,j}\vert\Big)$
$\lt 1$
ii.) $\vert r_{j,j}\vert = 1$ but $r_{j,j} \neq 1$
$\vert u_{j,j}\vert $
$= \frac{1}{n}\Big\vert r_{j,j}+r_{j,j}^2+....+ r_{j,j}^n\Big\vert $
$\lt \frac{1}{n}\Big(\big\vert r_{j,j}\vert +\vert r_{j,j}\vert ^2+....+ \vert r_{j,j}\vert^n\Big)$
$= \frac{1}{n}\Big(1+1+.... +1\Big)$
$= 1$
($\vert r_{j,j}\vert \gt 1$ is of course impossible)
so if you count exactly $m$ ones on the diagonal of $R$ (eigenvalue of T), then there are exactly $m$ ones on the diagonal of $U$ (eigenvalue of P).