Let $k$ be a field or a commutative ring with unit and let $F:M\_n(k)\to M\_n(k)$ be a $k$-linear map. Suppose that $F$ is given in the form $F(X) = A\_1XB\_1 + \cdots + A\_m X B\_m$ for some $A\_i,B\_i\in M\_n(k)$ (note that any $k$-linear $F$ can be written in this form, though not uniquely). 



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> Is there a formula that allows one to determine $\mathrm{det}(F)$ (no pun intended) or $\mathrm{tr}(F)$ directly from the matrices $A\_i, B\_i$ ($1\leq i\leq m$), i.e. without having to compute a representation matrix of $F$?
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Determinants (and traces) of linear maps of matrices
We have $F(X)=\sum\_i A\_i X B\_i = \sum\_i (B\_i^T \otimes A\_i) vec(X)$ (see [here](https://en.wikipedia.org/wiki/Vectorization_(mathematics))), i.e. $F \sim \sum\_i (B\_i^T \otimes A\_i)$. Because of some formulas [here](https://en.wikipedia.org/wiki/Kronecker_product) we have $tr(F)=\sum\_i tr(A\_i)tr(B\_i)$. For the determinant I don't think there is a nice formula.


