Invertibility over rings

1. A.If A is a real 3×2 matrix and B a real 2×3 matrix with BA $= I_{2}$, then AB $= I_{3}$.
2. B.Let A = [[3, 3], [1, 2]]. Then there is a matrix B with integer entries such that AB $= I_{2}$.
3. C.Let A = [[3, 1], [1, 2]] with entries in $\mathbb{Z}/6\mathbb{Z}$. Then there is a matrix B with entries in $\mathbb{Z}/6\mathbb{Z}$ such that AB $= I_{2}$.✓ Correct
4. D.If A is a real non-zero 3×3 diagonal matrix, then there is a real matrix B such that AB $= I_{3}$.

The trap it tests

Something true in ℝⁿ, or in a nice space, assumed in general.