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We solve for quantum-geometrically realised spectral triples or `Diracoperators' on the noncommutative torus $\Bbb C_\theta[T^2]$ and on the algebra$M_2(\Bbb C)$ of $2\times 2$ matrices with their standard quantum metrics andassociated quantum Levi-Civita connections. For $\Bbb C_\theta[T^2]$, we obtainan even standard spectral triple but now uniquely determined by full geometricrealisability. For $M_2(\Bbb C)$, we are forced to the flat quantum Levi-Civitaconnection and again obtain a natural fully geometrically realised evenspectral triple. In both case there is also an odd spectral triple for adifferent choice of a sign parameter. We also consider an alternate quantummetric on $M_2(\Bbb C)$ with curved quantum Levi-Civita connection and find anatural 2-parameter of almost spectral triple in that $D$ fails to beantihermitian. In all cases, we split the construction into a local tensoriallevel related to the quantum geometry, where we classify the results morebroadly, and the further requirements relating to the Hilbert space structure.We also illustrate the Lichnerowicz formula for $D^2$ which applies in the caseof a full geometric realisation.

Geometric Dirac operator on noncommutative torus and $M_2(\Bbb C)$