Extremal Matrix Problems for Lucas Numbers and a Bilateral Fibonacci–Lucas Sum

Using only the companion matrix trace, we solve two recently proposed problems on Lucas and Fibonacci numbers. The first is an inequality involving even-index Lucas numbers; we sharpen it to the exact lower envelope for each fixed parameter, thereby determining all equality cases. The second is an infinite bilateral reciprocal series; we embed it into a universal summation identity valid for any real base with absolute value less than one, and then specialize to obtain the claimed closed form. The proofs are self-contained, including convergence estimates and negative-index conventions. The method also suggests two broader open questions about generalized trace envelopes and rational telescoping classifications.