Linear Algebra and rank problem.

[georgeh]

I have the following problem which I can't figure out.

Let A = [a_11,a_12;a_13; a_21; a_22; a_23;]
Show that A has rank 2 if and only if one or more of the determinants

| a_11,_a_12; a_21,a_22| , |a_11,a_13;a_21,a_23|,|a_12,a_13;a_22,a_23|
I know its a 2x3 matrix..which the det. wouldn't apply since it is not square. Not sure how to proceede

 

[HallsofIvy]

You might start by stating the problem correctly

"Show that A has rank 2 if and only if one or more of the determinants
| a_11,_a_12; a_21,a_22| , |a_11,a_13;a_21,a_23|,|a_12,a_13;a_22,a_23|" is what?? What is supposed to be true about them?

"I know its a 2x3 matrix..which the det. wouldn't apply since it is not square."
That's irrelevant- the problem doesn't say anything about the determinant of A (which doesn't exist) only the determinants of those 2 by 2 subsets.

 

[georgeh]

You might start by stating the problem correctly

"Show that A has rank 2 if and only if one or more of the determinants
| a_11,_a_12; a_21,a_22| , |a_11,a_13;a_21,a_23|,|a_12,a_13;a_22,a_23|" is what?? What is supposed to be true about them?
the det is not equal to zero..
"I know its a 2x3 matrix..which the det. wouldn't apply since it is not square."
That's irrelevant- the problem doesn't say anything about the determinant of A (which doesn't exist) only the determinants of those 2 by 2 subsets.

Yeah i states how they had it though. sorry.
they said.. the det != 0

 

[HallsofIvy]

Okay, now that we have that straightened out, exactly what is your definition of "rank"? What do you get if you "row reduce" A?

 

[georgeh]

the rank is the dimensions of the row space and column space of a matrix.
So when we do r-r-e, wherever we get leading ones, that is the rank #.