Is this observation regarding matrices correct?

Question

If a $ 2$ by $2$ matrix has a determinant of $0$ then the endpoints of the column vectors point away from the origin in the same direction and endpoints of the row vectors point away from the origin in the same direction $ \Bbb{R^2}$

If a $ 3$ by $3$ matrix has a determinant of $0$ then the endpoints of the column vectors are on the same line in $ \Bbb{R^3}$ and the endpoints of the row vectors are on the same line in $ \Bbb{R^3}$

Answer 1

Depends on what you mean by "point away" (consider the zero matrix, which consists of zero vectors, all "pointing" in arbitrary directions).

Otherwise, you are partially correct. In general, if an $n$ by $n$ determinant is 0, it simply means that the column vectors do not span Rⁿ. This does not necessarily mean that they need to meet in one line.

Let us look at the 3 by 3 case. We do not need to consider the row vectors, since we know they span a vector space with the same dimension as the column vectors. So, if the column vectors span a plane, then the row vectors do so as well.

Now, if the determinant is 0, this could mean one of the following:

• Two column vectors are parallel (and can be placed on the same line) and the third one, together with one of the parallel vectors, span a plane in R³.
• All three column vectors are parallel, and can be placed on a line, as you suggested. In this case, they span a line.

Hope this helped.