Using the determinantal representation and conditions for the existence of the Drazin inverse from the paper of Stanimirović and Djordjević (Linear Algebra Appl. 311 (2000), 131-151), we introduce a determinantal representation of the minimal $P$-norm solution of a given linear system. More precisely, we represent elements of the minimal $P$-norm solution $A^{\mathcal D}b$ as fractions of two expressions involving minors of the order $\operatorname{rank}(A^k)=\operatorname{ind}(A)$, taken from the matrix $A$ and its rank invariant powers $A^l$, $l\geq k$.