Matrix theory holds significant importance for the resolution of certain real-world problems and computational processes. Specific types of matrices and their lin-ear algebraic properties are of considerable significance. In this paper, we analyse the main features of the r-min and r-max matrices with harmonic higher order Gauss Fibonacci numbers entries, namely the inverse, determinant, permanent, and norms. Several recurrence relations for the characteristic polynomials will be also established. To verify our results, we present, as a pertinent application, the afore-mentioned properties of the r-min matrix, whose entries are harmonic higher order Gauss Fibonacci numbers.