Determining values for an vector entry to make vectors independent. Determine all values of $h$ such that the vector set $\{ (3, 1, 2), (0, 1, -1), (3, 3, h) \}$ is independent.
 A: Set up a matrix whose columns represent your vectors. (They could each represent a row in the matrix, instead). You need to find where the determinant of the matrix is equal to zero. Why?: That is when the vectors will NOT be linearly independent.
So we compute:
$$\left|\begin{matrix}
3 & 0 & 3 \\
1 & 1 & 3 \\
2 & -1 & h \\
\end{matrix}\right|
= 3h$$
$$3h = 0 \implies h=0$$
Since the determinant of the matrix is zero if and only if $h = 0$, the column vectors are linearly independent for all values $h \neq 0$.

Another way of seeing this is as follows:
$$\begin{pmatrix}
3 & 0 & 3 \\
1 & 1 & 3 \\
2 & -1 & h \\
\end{pmatrix}
\overset{\large R3 + R2 \to R2}{=}
\begin{pmatrix}
3 & 0 & 3 \\
3 & 0 & 3+h \\
2 & -1 & h \\
\end{pmatrix}
$$
What is the only value of $h$ that will force the second row to be row equivalent to the first row? (At what value of $h$ could you subtract row two from row one and up with a row of all zeros?) At that value of $h$, the rows are linearly dependent. The determinant of a matrix whose rows (columns) are linearly dependent is always zero. Hence can your vectors be linearly independent at that value of $h$? 
A: Consider the matrix $\left(\begin{matrix}3 & 1&2\\0&1&-1\\3& 3&h\end{matrix}\right)$. This has determinant $3h$. Now think about what the determinant tells you about independence of the row vectors.
