Assume that $X\sim \operatorname{IW}_p(n,\Sigma)$ has an inverse Wishart distribution, which probability density function is $$f(X\mid n,\Sigma)=C(n,\Sigma)|X|^{-\frac{n+p+1}{2}} \exp\Big(-\frac{1}{2} \operatorname{tr}(\Sigma X^{-1})\Big), \tag 1$$ 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 $X_{11}\sim \operatorname{IW}(n-p_2,\Sigma_{11})$, by the probability density function of $X$ in (1) directly, not the definition of $\operatorname{IW}$ or Wishart distribution???

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