On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix

We characterize the involutiveness of linear combinations form $a{\mathbf{A}} + b{\mathbf{B}}$ when $a,b$ are nonzero complex numbers, ${\mathbf{A}}$ is a quadratic $n \times n$ matrix and ${\mathbf{B}}$ an arbitrary matrix, under certain properties imposed on $\mathbf{A}$ $\mathbf{B}$.