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:
$Ax + By + Cz = 0_{3 \times 1}$
Suppose that I would like to prove some result involving $A$, $B$, and $C$ given the above equation. Would it be enough to prove the result separately in the following cases?
Case 1: $Ax=0_{3 \times 1}$, $By=0_{3 \times 1}$, and $Cz=0_{3 \times 1}$.
Case 2: $By+Cz=0_{3 \times 1}$.
Case 3: $Ax+Cz=0_{3 \times 1}$.
Case 4: $Ax+By=0_{3 \times 1}$.
Case 5: $By+Cz\neq0_{3 \times 1}$.
Case 6: $Ax+Cz\neq0_{3 \times 1}$.
Case 7: $Ax+By\neq0_{3 \times 1}$.