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 $=I_2$.
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

Finite-dimensional intuition

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