For what values of h will $\{(1,2,3),(2,-1,4),(3,h,4)\}$ be linearly independent?
This is how I set up my matrix:
$$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & -1 & h \\ 3 & 4 & 4\end{pmatrix}$$
After row reducing it, this is the matrix I ended up with:
$$A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & -5 & h-6 \\ 0 & 0 & -2\end{pmatrix}$$
I don't know what to do from here.
Should I have switched $R_2$ with $R_3$ in the original matrix $A$ in order to get the $h$ in position for the last pivot and then rref to answer the question?
Best Answer
See if you can get the matrix to row reduced form
$$\begin{pmatrix} 1 & 0 & -2 \\ 0 & 1 & 5/2 \\ 0 & 0 & h+13/2 \end{pmatrix} $$
and presumably you know that if the three vector are linearly independent the matrix in RREF must satisfy certain condition.