Question

Hey guys, help me with this.
Let A be an m x n matrix and let c be a vector in Rm.
Then the system of equations Ax = c has a unique solution if and only if rank (A) = rank (Ac) = n.
Prove.

Answer 1

well the rank is the number of pivots.. if the rank of the matrix does not =n then there can't be a unique solution.. There would either be 0 or infinitely many

Answer 2

in other words the rank has to equal the number of variables