Consider a system of linear equations with augmented matrix A and coefficient matrix C.
In each case either prove the statement or give an example showing that it is false.
(a) If there is more than one solution, A has a row of zeros.
(b) I f A has a row of zeros, there is more than one solution.
(c) If there is no solution, the row-echelon form of C has a row of zeros.
(d) If the row-echelon form of C has a row of zeros, there is no solution.
(e) There is no system that is inconsistent for every choice of constants.
(f) If the system is consistent for some choice of constants, it is consistent for every choice of constants.
Now assume that the augmented matrix A has 3 rows and 5 columns.
(g) If the system is consistent, there is more than one solution.
(h) The rank of A is at most 3.
(i) If rank A = 3, the system is consistent.
(j) If rank C= 3, the system is consistent.