Determinant of the sum of rank-$1$ matrices

Question

Let $A, B$ be a $n \times n$ diagonal matrices and $C = c c^T$ be also an $n \times n$ matrix. What is the determinant of $ACB + BCA$? Can we represent the determinant using $\det A$, $\det B$ and $\det C$?

Answer 1

$ACB+BCA$ is the sum of two rank-1 matrices. So, its rank is at most two and the determinant is always zero when $n\ge3$. When $n=2$, the matrix is given by $M=u v^T + v u^T = [u|v][v|u]^T$, where $u=Ac,\ v=Bc$ and $[u|v]$ denotes an augmented matrix. Hence $$ \det M=\det([u|v][v|u]^T)=\det([v|u]^T[u|v])=(u^Tv)^2-(u^Tu)(v^Tv). $$ This has very little to do with the determinants of $A,B$ and $C$.