Suppose $X\sim IW_{p}(n,\Sigma)$ has an inverse Wishart distribution, which probability density function is $$f(X\mid n,\Sigma)=C(n,\Sigma)|X|^{-\frac{n+p+1}{2}} \exp\Big(-\frac{1}{2} \operatorname{tr}(\Sigma X^{-1})\Big),~~\qquad (1)$$ where $C(n,\Sigma)$ is a constant, only depends on $n$ and $\Sigma.$
Partition the matrices $X$ and $\Sigma$ conformably with each other $$X=\pmatrix{X_{11}&X_{12}\\X_{21}&X_{22}},~~\Sigma=\pmatrix{\Sigma_{11}&\Sigma_{12}\\\Sigma_{21}&\Sigma_{22}}.$$ where $X_{ij}$ and $\Sigma_{ij}$ are $p_i\times p_j$ matrices.
How to prove $X_{11}\sim IW(n-p_2,\Sigma_{11})$, by the probability density function of $X$ in (1) directly, not the definition of $IW$ or Wishart distribution???
