Let $G_{n,k}({\mathbb K})$ be the Grassmannian manifold of $k$ -dimensional ${\mathbb K}$ -subspaces in ${\mathbb K}^n$ , where ${\mathbb K}={\mathbb R}, {\mathbb C}, {\mathbb H}$ is the field of real, complex, or quaternionic numbers. For $1\le k \lt k^\prime \le n-1$ , we define the Radon transform $({\mathcal R}f)(\eta)$ , $\eta \in G_{n,k^{\prime}}({\mathbb K})$ , for functions $f(\xi)$ on $G_{n,k}({\mathbb K})$ as an integration over all $\xi \subset \eta$ . When $k+k^\prime \le n$ , we give an inversion formula in terms of the Gårding-Gindikin fractional integration and the Cayley-type differential operator on the symmetric cone of positive ( $k\times k$ )-matrices over ${\mathbb K}$ . This generalizes the recent results of Grinberg and Rubin [4] for real Grassmannians