How to determine whether there is a linear dependence between rows or columns in a matrix? When given a matrix, such as the one below, what should I do to determine whether there is a linear dependence between rows or columns?
For example:
2    -1     4
8     6     0
3     1     2

Given no other sources of information. 
 A: You can calculate the determinant: $=2\cdot 6\cdot 2+4\cdot 8-3\cdot 4\cdot 6+2\cdot 8=0$.
If the determinant is $0$, then the rows or columns are dependent.
A: If the determinant of a square matrix is zero, there is a dependence.
In a rectangular matrix there is a dependence between the rows if there are more rows than columns, and between the columns if there are more columns than rows.
If there are more rows than columns, it is less straightforward to detect a dependence between the columns. You can take the square matrix comprised of the first rows and all the columns, and if the determinant is zero, there may be a dependence (the dependence in the square matrix has to extend consistently - but the dependence in the top square may not be unique, so all dependences have to be tested).
A: If you take the determinent of the matrix, if it is zero there is a liner dependency among the rows (and among the columns).  
IN this case, the determinent is 42, so no linear combination of rows with coefficients other than all zero come out to be zero. 
