Let $A,B,C \in M_{3 \times 3}(\mathbb{R})$ be fixed and let $x,y,z \in M_{3 \times 1}(\mathbb{R})$ be any three column vectors. Consider the matrix equation given by
$Ax + By + Cz = 0_{3 \times 1}$
All multiplication of matrices is the usual row-column definition. I am interested in when the following three relations simultaneously hold:
$Ax = 0_{3 \times 1}$
$By = 0_{3 \times 1}$
$Cz = 0_{3 \times 1}$
That is, what property or properties must $A$,$B$, and $C$ possess in order to make this so? I thought about this for a while, but haven't really come to a very good conclusion yet. Any ideas?