how to find all triples $(c_1, c_2, c_3)?$

Question

Let $$ A= \begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23} \end{bmatrix} $$ be a real matrices . let $c_1 , c_2, $and$ c_3 $ be the scalars

$$ c_1 = det( \begin{bmatrix} a_{12}&a_{13}\\ a_{22}&a_{23} \end{bmatrix}) $$ $$ c_2 = det( \begin{bmatrix} a_{13}&a_{11}\\ a_{23}&a_{21} \end{bmatrix}) $$ $$ c_3 = det( \begin{bmatrix} a_{11}&a_{12}\\ a_{21}&a_{22} \end{bmatrix}) $$

Find all triples $(c_1, c_2, c_3) $so that $rank A = 2$

this is the orginal question

i was looking this problem at book linear algebra by Kennneth hoffman and Ray kunze (2 edition pearson publisher) page no : 149 Q.10

It is given that $rank(A) = 2$ if and only if ($c_1,c_2, c_3$) $\neq 0$

here i don't know how can i find all the triples $(c_1, c_2, c_3) $so that $rank A = 2$

i would be thankful if someone help me,,,,

thanks in advance

Answer 1

HINT

Note that to have rank=2 we need two column linearly independent; the third can be

1. independent from each one of the others or
2. can be a multiple of one of them