Exercise 1.31 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 can- not be homogeneous. C. If there exists a trivial solution, the system is homogeneous. d. If the system is consistent, it must be homo- geneous. Now assume that the system is homogeneous. e. If there exists a nontrivial solution, there is no trivial solution.

a. False. A homogeneous system always has at least the trivial solution, where all variables are equal to zero. However, it can have nontrivial solutions as well. For example, the system x + y = 0 has the trivial solution x = 0 and y = 0, but also has the nontrivial solution x = 1 and y = -1.<br /><br />b. True. A system of linear equations is homogeneous if the constant term in each equation is zero. If a system has a nontrivial solution, it means that there is at least one equation in the system that is not satisfied by the trivial solution (all variables equal to zero). Therefore, the system cannot be homogeneous.<br /><br />c. False. A system of linear equations is homogeneous if the constant term in each equation is zero. If there exists a trivial solution, it means that the system is homogeneous, but it can also have nontrivial solutions.<br /><br />d. False. A system of linear equations is consistent if it has at least one solution. It can be either homogeneous or non-homogeneous. For example, the system x + y = 2 is consistent but not homogeneous.<br /><br />e. False. If a system is homogeneous, it can have both trivial and nontrivial solutions. The trivial solution is always present, but the presence of a nontrivial solution does not mean that there is no trivial solution.