I've run into the following expression involving the permanent of a matrix: $$ \operatorname{per}\left( \left[ \begin{array}{cc} A & B\\ I & C \end{array} \right] \right) $$ Where $A$ and $C$ are diagonal matrices, $B$ is a dense matrix and $I$ is the identity matrix. All of them are $n\times n$ matrices.
I'm not very familiar with the properties of the permanent, but can in this case be expressed in terms of the blocks $A, B$ and $C$? As far as I've read, there is no general rule for the permanent of a block matrix in terms of the different blocks, but since this is a particular case (with diagonals matrices) there may be a formula.
I think this should hold: $$ \operatorname{per}\left( \left[ \begin{array}{cc} A & B\\ I & C \end{array} \right] \right) = \operatorname{per}\left( AC +B \right) $$
What if $C$ is not diagonal?