Answer Happy

7. (10pt) Either prove the following statements, or give an example to show that it is false. (a) If a system of m linear equations in n variables has coefficient matrix C (an m x n matrix) and augmented matrix A with m <n, then the system has infinitely many solutions. (b) If a consistent system of m linear equations in n variables has coefficient matrix C (an m x n matrix) and augmented matrix A with m <n, then the system has infinitely many solutions. (c) If a system of m linear equations in n variables has coefficient matrix C (an mx n matrix) and augmented matrix A with the rank of A greater than the rank of C, then the system is inconsistent. (d) If a homogeneous system of m linear equations in n variables has coefficient matrix C (an m x n matrix) and mugmented matrix A where the rank of A is less than n, then the system has non-trivial solutions. (e) If a sequence of row operations is done to a consistent system of linear equations, then the resulting system is also consistent