Problem
Prove $ \ \ \begin{vmatrix} a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \\ b_{1} & b_{2} & b_{3} & b_{4} & b_{5} \\ c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ d_{1} & d_{2} & 0 & 0 & 0 \\ e_{1} & e_{2} & 0 & 0 & 0 \\ \end{vmatrix} = 0. $
Restrictions: All variables are nonzero.
Attempt
My first attempt was to reduce the bottom two rows to zero, and another attempt was to transform the matrix into a lower triangular matrix. Neither worked.
Notes
If anyone can provide me detailed steps on how to achieve the RHS, I'd be very appreciative! Thanks!
Hint By Row Operations you can reduce the bottom two rows
$$\begin{vmatrix} a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \\ b_{1} & b_{2} & b_{3} & b_{4} & b_{5} \\ c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ d_{1} & d_{2} & 0 & 0 & 0 \\ e_{1} & e_{2} & 0 & 0 & 0 \\ \end{vmatrix} = \begin{vmatrix} a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \\ b_{1} & b_{2} & b_{3} & b_{4} & b_{5} \\ c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ 1 & 0 & 0 & 0 & 0 \\ 0 & d_1e_{2}-d_2e_2 & 0 & 0 & 0 \\ \end{vmatrix} $$ Then, by expnasion you get $$\begin{vmatrix} a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \\ b_{1} & b_{2} & b_{3} & b_{4} & b_{5} \\ c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ d_{1} & d_{2} & 0 & 0 & 0 \\ e_{1} & e_{2} & 0 & 0 & 0 \\ \end{vmatrix} = \left(d_1e_{2}-d_2e_2\right)\begin{vmatrix} a_{3} & a_{4} & a_{5} \\ b_{3} & b_{4} & b_{5} \\ c_{3} & c_{4} & c_{5} \\ \end{vmatrix} $$
which can be calculated explicitely and is not identically 0.