Consider the following statements about a system of linear equations with augmented matrix A. In each case either prove the statement or give an example for which it is false.
(a) If the system is homogeneous, every solution is trivial.
(b) If the system has a nontrivial solution, it cannot be homogeneous.
(c) If there exists a trivial solution, the system is homogeneous.
(d) If the system is consistent, it must be homogeneous.
Now assume that the system is homogeneous.
(e) If there exists a nontrivial solution, there is no trivial solution.
(f) If there exists a solution, there are infinitely many solutions.
(g) If there exist nontrivial solutions, the row- echelon form of A has a row of zeros.
(h) If the row-echelon form of A has a row of zeros, there exist nontrivial solutions.
(i) If a row operation is applied to the system, the new system is also homogeneous.