Explicit Formulas for the Anti-Trace of 3-by-3 Matrix Powers via Generating Functions

Jeriko Gormantara (Department of Mathematics, Universitas Hasanuddin)
Devi Fitri Ferdania (School of Mathematics, University of Birmingham)
Fajar Yuliawan (Department of Mathematics, Institut Teknologi Bandung)
Hanni Garminia (Department of Mathematics, Institut Teknologi Bandung)

Journal of the Indonesian Mathematical Society
Vol. 32 No. 3 (2026): SEPTEMBER

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.