Singularities of determinantal pure pairs

Let $X$ be a generic determinantal affine variety over a perfect field of characteristic $p\ge 0$ and $P\subset X$ be a standard prime divisor generator of $\mathrm{C}\mathrm{l}(X)\cong \mathbb{Z}$. We prove that the pair $(X,P)$ is purely $F$-regular if $p>0$ and so that $(X,P)$ is purely log terminal (PLT) if $p=0$ and $(X,P)$ is log $\mathbb{Q}$-Gorenstein. In general, using recent results of Z. Zhuang and S. Lyu, we show that $(X,P)$ is of PLT-type, i.e. there is a $\mathbb{Q}$-divisor $\mathrm{\Delta }$ with coefficients in $[0,1)$ such that $(X,P+\mathrm{\Delta })$ is PLT.