I'm asking this just out of curiosity because a brief googling failed to give me the answer.
By skew-triangular matrices I mean matrices with this $ \begin{bmatrix} \times & \times & \times & \times \\ \times & \times & \times \\ \times & \times \\ \times \\ \end{bmatrix} $ or this $ \begin{bmatrix} & & & \times \\ & & \times & \times \\ & \times & \times & \times \\ \times & \times & \times & \times \\ \end{bmatrix} $ sparsity pattern. Simple experiments with Mathematica show that the inverse of the first type is a matrix of the second type (and vice versa, of course).
1. Do these matrices have their real name and where do they occur?
2. What other interesting properties do they possess?