Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$ \det(I+AB) = \det(I+BA), $$ where in the first case $I$ denotes the $m\times m$ identity, and in the second, the $n\times n$ identity. Could you sketch a proof for me, or point ...