Rank $2$ matrices

Question

Question is to show that the set of rank $2$ matrices in $M_{2\times 3}(\mathbb{R})$ is open...

Let $M$ be the set of rank $2$ matrices in $M_{2\times 3}(\mathbb{R})$..

Let $N$ be the complement of $M$... Then $B$ consists of rank $1$ matrices with zero matrix..

We prove that $B$ is closed..

Any rank $1$ matric is of the form $\begin{bmatrix}a&b&c\\a&b&c\end{bmatrix}$ or $\begin{bmatrix}a&a&a\\b&b&b\end{bmatrix}$ with atleast one of $a,b,c$ non zero..

Let $A_n=\begin{bmatrix}a_n&b_n&c_n\\a_n&b_n&c_n\end{bmatrix}\rightarrow B=\begin{bmatrix}a&b&c\\a&b&c\end{bmatrix}$

Let $A_n=\begin{bmatrix}a_n&a_n&a_n\\b_n&b_n&b_n\end{bmatrix}\rightarrow B=\begin{bmatrix}a&a&a\\b&b&b\end{bmatrix}$ This clearly says $B$ is closed.. The point is the limit may be zero that is why i have included zero matrix also..

So, I want to know if there are any gaps..

I was thinking of some other argument...Could not succeed..

We have $rank(A)=rank(AA^T)$.. Consider $\det:M_{2\times 3}(\mathbb{R})\rightarrow \mathbb{R}$ with $A\rightarrow \det(AA^T)$

Then $\det^{-1}(\{0\})=\{A:rank(AA^T)\neq 2 \}=\{A:rank(A)\neq 2 \}$ is closed...

So, $\{A:rank(AA^T)= 2 \}$ is open...

See if there are any gaps...

Answer 1

Hint For aan alternate solution, you might define $$d_1 (\begin{bmatrix} a_{11} & a_{12} & a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix})= \det(\begin{bmatrix}a_{11} & a_{12}\\a_{21}&a_{22}\end{bmatrix}) \\ d_2(\begin{bmatrix} a_{11} & a_{12} & a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix})= \det(\begin{bmatrix}a_{11} & a_{13}\\a_{21}&a_{23}\end{bmatrix})\\ d_3 (\begin{bmatrix} a_{11} & a_{12} & a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix})= \det(\begin{bmatrix}a_{12} & a_{13}\\a_{22}&a_{23}\end{bmatrix})$$

Now, these functions are continuous, and the set you are trying to prove is close is $d_1^{-1}(0) \cap d_2^{-1}(0) \cap d_3^{-1}(0)$

Answer 2

Your second argument is correct, but could be phrased more clearly.

You've made a mistake with the first. What you should say is that any rank $1$ matrix has the form $$ \pmatrix{a_1 \\ a_2}\pmatrix{b_1&b_2 & b_3} = \pmatrix{ a_1b_1 & a_1b_2 & a_1b_3\\ a_2b_1 & a_2b_2 & a_2b_3 } $$