Determinant of specially structured block matrix

It has the same determinant as the matrix $$ \begin{pmatrix} B & 0 & \cdots & -B \\ 0 & B & \cdots & -B \\ \vdots & \vdots & \ddots & \vdots\\ A & A & \cdots & A+B \end{pmatrix}, $$ which has the same determinant as $$ \begin{pmatrix} B & 0 & \cdots & 0 \\ 0 & B & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ A & A & \cdots & nA+B \end{pmatrix}, $$ which is easy to compute.