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Devi Fitri Ferdania
School of Mathematics, University of Birmingham
Mathematics

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Explicit Formulas for the Anti-Trace of 3-by-3 Matrix Powers via Generating Functions Jeriko Gormantara; Devi Fitri Ferdania; Fajar Yuliawan; Hanni Garminia
Journal of the Indonesian Mathematical Society Vol. 32 No. 3 (2026): SEPTEMBER
Publisher : IndoMS

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.22342/jims.v32i3.2260

Abstract

We study the anti-trace—the sum of anti-diagonal entries—of powers of 3-by-3 matrices. Unlike the ordinary trace, the anti-trace has no spectral characterization, so closed forms for the anti-trace of matrix powers are valuable. Starting from the Cayley--Hamilton recurrence for matrix powers and solving a trivariate generating function, we derive explicit closed-form expressions whose coefficients are signed multinomial combinations of the sums of principal minors and their anti-principal counterparts. The formulas are purely algebraic and remain valid over any commutative ring. As a structural application, we analyze block anti-diagonal matrices of size 3m-by-3m.
Co-Authors Fajar Yuliawan Hanni Garminia Jeriko Gormantara