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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?

user1551
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JSCB
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1 Answers1

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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 $\det(AB)=0$.

Ángel Mario Gallegos
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  • And the same is true in general. If the product of two nonsquare matrices gives a square matrix it will never be invertible. – leo May 01 '14 at 07:01
  • Learning little facts like this is why I go on mathSE. – Caleb Stanford May 01 '14 at 07:37
  • @leo That's false, since you can find two vectors whose dot product is nonzero. – Nishant Aug 06 '14 at 04:43
  • @Nishant you're right, let me correct myself: Let $A$ be a $m\times n$ matrix and $B$ be a $n\times m$ matrix. If $n\lt m$ then $AB$ is not invertible. – leo Aug 06 '14 at 04:58