echelon form
a matrix is in echelon form if
- in every row, the first nonzero entry is 1, called a pivotal 1.
- the pivotal 1 of a lower row is always to the right of the pivotal 1 of a higher row.
- in every column that contains a pivotal 1, all other entries are 0.
- any rows consisting entirely of 0's are at the bottom.
pivotal variables correspond to the rows that have a pivotal 1 (from echelon_form.html), and non-pivotal variables are the free variables.
once we reduce (convert progressively) a matrix to its echelon form, we would be able to clearly see how many solutions said matrix has
a matrix with no solution has the following echelon shape:
its not possible that 
so such a matrix has no solution
a matrix with no solution has the following echelon shape:
its not possible that so such a matrix has no solutiona matrix with a single solution has the following echelon shape:
this is the kind of shape we would hope for when reducing a matrix to its echelon form, because the solution is very easy to read which is:
this is the kind of shape we would hope for when reducing a matrix to its echelon form, because the solution is very easy to read which is:a matrix with an 
number of solutions has the following echelon shape:
where one of the rows zeros out which makes the third variable in the corresponding system of equations a free variable that can take any value, the other 2 variables depend on the value of the free variable
number of solutions has the following echelon shape:where one of the rows zeros out which makes the third variable in the corresponding system of equations a free variable that can take any value, the other 2 variables depend on the value of the free variable