The equiprime, strongly equiprime and uniformly strongly equiprime radicals were recently introduced by the present authors, together with S. Veldsman. We show that, for a near-ring R, \({\mathcal R}(R)^*={\mathcal R}({\mathcal M}_ n(R))\) where \({\mathcal M}_ n(R)\) is the matrix near-ring as defined by Meldrum and Van der Walt and \({\mathcal R}\) denotes the equiprime or strongly equiprime radical. An inclusion is also shown in the case of the uniformly strongly equiprime radical.