I have this matrix here $$ \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ r & -4 & -6 \\ \end{vmatrix} $$
with unknown value of $r$. I want to find this value so that $AX=0$ is true. What I have done so far is set up $3$ equations:
$$x_1+2x_2+3x_3=0$$
$$4x_1+5x_2+6x_3=0$$
$$rx_1-4x_2-6x_3=0$$
At this point, I am unsure how to proceed, as if I continue following Cramer's rule and set up determinants, I would get $0$ to be values for $x_1,x_2,x_3$.
You are correct.
If the determinat is not zero the only solution to the homogeneous system is $(0,0,0)$ If you make the determinant $0$ then you have non zero solutions as well. Is there a value for $r$ to make determinant $0$?