Is there a general formula for determining the number of free variables in a square matrix and a non-square one?
Must I row-reduce these matrices before determining the number or can I directly figure it out?
I have some intuition with square matrices. Basically, if I row-reduce a square matrix to its reduced echelon form, then the number of free variables is equal to the number of zero rows correct? Does it have to be RREF or can it just be REF?
SUMMARY
I am just wondering how I can determine the number of free-variables in both types of matrices and if it can be done by just looking at the matrix or if some sort of reduction must be done.
Best Answer
Reducing the matrix to row-echelon form suffices as you can see which columns are pivot and non=pivot.
If it is a square matrix, the number of non-pivot columns is equal to the number of zero rows.
However, if the matrix is non-square, you can reduce to row-echelon form and count the number of non-pivot columns. Here, the number of non-pivot columns is not equal to the number of zero rows.
Here there are 2 non-pivot columns and 1 zero row:
$\left(\begin{array}{}1&2&3\\ 0&0&0\end{array}\right)$