Let $B$ be a fixed $n\times n$ matrix. Find the determinant of the following matrix:
\begin{align} \begin{pmatrix} B-B_{11}I&-B_{21}I&\cdots&-B_{n1}I\\ -B_{12}I&B-B_{22}I&\cdots&-B_{n2}I\\ \vdots&\vdots&\ddots&\vdots\\ -B_{1n}I&-B_{2n}I&\cdots&B-B_{nn}I \end{pmatrix}. \end{align}
I obtained the matrix because I wanted to calculate the determinant of the linear operator $T_B(A) = BA - AB$ directly from the matrix. I know that the determinant must be equal to $0$. Are there any methods to calculate this determinant without using arguments like singularity?