  The well known formula of B. Fuglede expresses the mean value of the Radon
k-plane transform on $R^n$ as a Riesz potential. We extend this formula to the
space of $n \times m$ real matrices and show that the corresponding matrix
k-plane transform $f \to \hat f$ is injective if and only if $n-k \ge m$.
Different inversion formulas for this transform are obtained. We assume that $f
\in L^p$ or $f$ is a continuous function satisfying certain "minimal"
conditions at infinity.
