There is a lemma in hoffman Kunze which states that:
If A is a $m \times n $ matrix where $m<n$ then it has a non-trivial solution.
I understand the proof of this which uses the fact that the number of non-zero rows is less than the number of unknowns.
However, my problem lies in the fact that if $\det(A)=0$ then $AX=0$ must not necessarily have a non-trivial solution.
Example:If $A$ is a $3 \times 3$ matrix then $AX=0$ represents $3$ planes. Assume that $2$ of them lie on each other while the third is incident at an angle different from the two such that it intersects only at the point $\{(0,0,0)\}$ with the other two.In such a case,$AX=0$ can only have a trivial solution but the number of non-zero rows is less than the number of unknowns.
Can someone point out my mistake.