The below is just me trying to explain my thought process, but it's not necessary to read it to understand my question. My question is just: what does it mean when the rank of a matrix is MORE than the number of variables in the matrix?
Let's say A is an augmented matrix in reduced row echelon form. I know that:
If the rank of A is less than the number of variables in the system of linear equations that A is representing, then there are an infinite amount of solutions to the system (there are x free variables where x = number of variables-rank[A]). The way I justify this is: if there isn't at least one row per variable, at least one variables is "unaccounted for" and therefore can have an infinite amount of values.
If the rank of A is equal to the number of variables (i.e. x = 0 using the above definition of x), then there is a single, unique solution).
But what does it mean when there are MORE non-zero rows than variables? Using the above though process, this would mean that one variable is accounted for more than once. Does that mean there are no solutions to the matrix because one variable is said to have two different values? What if the values agree with each other?
Sorry if this is a stupid question, any help is greatly appreciated!