In this paper, we study a structured family of matrices whose entries are given by products of $k$-Fibonacci and $k$-Lucas numbers. For this family, we obtain explicit and unified formulas for several classical matrix invariants, including the determinant, inverse, trace, and matrix powers, revealing nontrivial algebraic patterns induced by the underlying recurrence relations. In addition, we determine the spectral radius and the energy of the graphs naturally associated with these matrices. Finally, we establish connections between the resulting formulas and certain integer sequences recorded in the On-Line Encyclopedia of Integer Sequences (OEIS).