How to prove that $\det\left[\pmatrix{u_1 & v_1\\ u_2 & v_2\\ u_3 & v_3}\pmatrix{s_1 & s_2 & s_3\\ t_1 & t_2 & t_3}\right]=0$?

Question

Evaluate $\det\left[\begin{pmatrix} u_1 & v_1\\ u_2 & v_2\\ u_3 & v_3 \end{pmatrix} \begin{pmatrix} s_1 & s_2 & s_3\\ t_1 & t_2 & t_3 \end{pmatrix}\right]$.

I really don't want to expand the matrix product, is there simpler way?

Answer 1

Let $A=\begin{pmatrix}u_1&v_1\\u_2&v_2\\u_3&v_3\end{pmatrix}$ and $B=\begin{pmatrix}s_1&s_2&s_3\\t_1&t_2&t_3\end{pmatrix}$, since $A$ is a $3\times 2$ matrix and $AB$ is a $3\times 3$ matrix it follows $\text{rank}(AB)\le\text{rank}(A)\le 2$, then $\det(AB)=0$.