Summary: We use the computer algebra system Maple to study the 512-dimensional associative algebra \(\mathbb Q \mathcal B_3\), the rational monoid algebra of \(3{\times}3\) Boolean matrices. Using the LLL algorithm for lattice basis reduction, we obtain a basis for the radical in bijection with the 42 non-regular elements of \(\mathcal B_3\). The center of the 470-dimensional semisimple quotient has dimension 14; we use a splitting algorithm to find a basis of orthogonal primitive idempotents. We show that the semisimple quotient is the direct sum of simple two-sided ideals isomorphic to matrix algebras \(M_d(\mathbb Q)\) for \(d=1,1,1,2,3,3,3,3,6,6,7,9,9,12\). We construct the irreducible representations of \(\mathcal B_3\) over \(\mathbb Q\) by calculating the representation matrices for a minimal set of generators.