While solving a question asking for the condition for 3 vectors to be linearly independent, I simple made a 3×3 column matrix and equated it to not 0 but what is the reason for doing so
, I guess matrix product will give me the cross product of all the vectors and if it's not 0 it means angle between them is not 0 but how does it tell they are not linearly dependent
A useful thing to keep in mind is that if $A=\begin{bmatrix} a_1 & a_2 & \dots & a_n \end{bmatrix}$ then $Ax=x_1 a_1 + x_2 a_2 + \dots + x_n a_n$. Therefore, if you have a nontrivial solution to $Ax=0$, say $x$, then the entries of $x$ are coefficients in a linear dependence relation between the vectors $a_1,a_2,\dots,a_n$.
The cross product doesn't really have anything to do with it, even in 3D where it makes sense.