So i have studied the rank method in matrix and how we can use it to find solutions or determine consistency of a system of equations, but despite having a qualititative idea,, i am unable to clearly understand why at all the method works, as to how the ranks have something to do with consistency,
What i do understand however is that the row transformations work because it is equivalent to manipulating multiple equations by multiplying different coefficients and then adding and subtracting,, we just do it inside the matrix,
From this i have deduced that ,,, probably since the rank of a matrix depends on number of non zero rows in echelon form,,
Thus if there remains a zero row in echelon form , it means that the particular row could be reduced to 0 simply through adding or subtracting (after multiplying scalars) with other rows OR that
the equations corresponding to other rows are equivalent to this one
that the three equations are actually just two or whatever number of non zero rows,
IS my logic correct,, what are the holes in it? and can you please provide a pdf so i can understand with greater clarity?