If $A = \begin{bmatrix} 1 & 1 & 2 \\ 2 & 5 & 4 \\ 1 & 0 & 5 \end{bmatrix}$, then the determinant of $\left(A^{2026} - 11A^{2025} - 9A^{2023}\right)$ is equal to:}
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For any characteristic equation of a $3 \times 3$ matrix given by $\lambda^3 - S_1\lambda^2 + S_2\lambda - |A| = 0$, $S_1$ represents the trace ($\text{sum of diagonal entries}$) and $|A|$ represents the determinant. Identifying these invariants directly from the matrix saves valuable time during matrix polynomial reduction.