Determine if the system has a nontrivial solution.
$-3x_1+4x_2-8x_3=0 $
$-2x_1+5x_2+4x_3=0 $
I know that in order to find if a system has nontrivial solutions, all on the entries in a matrix row are all 0s. Only two equations are given despite having 3 variables. Could I write a matrix of this system with 3 rows with the first two rows being the 2 equations and the last row being all 0s (because no 3rd equation is given despite having a 3rd variable)?
The system of linear equations $Ax=0$ has non-trivial solution when the rank of the matrix $A$ is smaller than the number of unknowns. For your problem, adding a 3rd row or not, the rank of the matrix is still 2 and there are 3 unknowns, so there must be at least one non trivial solution.