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