Let $A$ be a $3\times3$ complex matrix and $B$ its transpose. Let $ a$ be a complex number such that $ a \neq1$ and $\det(A+a*B)=0$. Compute $\det(A+B)$ in terms of $a$ and $\det(A).$
I tried to use the polynomial expansion $ \det(A+xB)=\det A + q*x +w*x^2+ \det B*x^3 $ for any matrices $A,B$. Probably I should have found some relations between coefficients $q$ and $w $ beacuse $B$ is $A$ transposed, but got stuck.