In: Advanced Math
Suppose the system AX = B is consistent and A is a 6x3 matrix. Suppose the maximum number of linearly independent rows in A is 3. Discuss: Is the solution of the system unique?
Since the system is consistent, there is at least one solution, say . Let be any solution to , and let ; then
Thus, .
Since the maximum number of linearly independent rows in is , we know that is the row-rank of . Since row-rank and column-rank of a matrix are identical, we conclude that column-rank of is also . Thus, all the columns of are linearly independent. Since the vector is a linear combination of the columns of , namely, , where and are columns of , we conclude that implies . This implies
Thus, if are solutions to then ; hence, the system has a unique solution.