Mathematics

The commutator $AB - BA$ measures how much two matrices fail to commute—and this particular commutator has a special structure: it's antisymmetric with trace zero. The key insight is that the commutator of any two matrices always has trace zero (since $\text{tr}(AB) = \text{tr}(BA)$), so your target matrix automatically satisfies a necessary condition. To find $A$ and $B$, try simple matrices with a few nonzero entries and use the constraint that their commutator must produce exactly those off-diagonal $\pm 3$ entries; scalar multiples of the identity will never work because they commute with everything.

The commutator $AB - BA$ measures how much two matrices fail to commute—and this particular commutator has a special structure: it's antisymmetric with trace zero. The key insight is that the commutator of any two matrices always has trace zero (since $\text{tr}(AB) = \text{tr}(BA)$), so your target matrix automatically satisfies a necessary condition. To find $A$ and $B$, try simple matrices with a few nonzero entries and use the constraint that their commutator must produce exactly those off-diagonal $\pm 3$ entries; scalar multiples of the identity will never work because they commute with everything.