46 CHAPTER 2. SYSTEMS OF LINEAR EQUATIONS

consistent, is the solution unique?■ ∗ ∗ ∗ ∗ | ∗0 ■ ∗ ∗ 0 | ∗0 0 0 0 ■ | 00 0 0 0 ∗ | ■

4. Suppose a system of equations has fewer equations than variables. Must such a

system be consistent? If so, explain why and if not, give an example which is notconsistent.

5. If a system of equations has more equations than variables, can it have a solution? Ifso, give an example and if not, tell why not.

6. Find h such that

(2 h | 43 6 | 7

)is the augmented matrix of an inconsistent matrix.

7. Find h such that

(1 h | 32 4 | 6

)is the augmented matrix of a consistent matrix.

8. Find h such that

(1 1 | 43 h | 12

)is the augmented matrix of a consistent matrix.

9. Choose h and k such that the augmented matrix shown has one solution. Then chooseh and k such that the system has no solutions. Finally, choose h and k such that the

system has infinitely many solutions.

(1 h | 22 4 | k

).

10. Choose h and k such that the augmented matrix shown has one solution. Then chooseh and k such that the system has no solutions. Finally, choose h and k such that the

system has infinitely many solutions.

(1 2 | 22 h | k

).

11. Find the solution in Z5 to the following system of equations.

x+2y+ z−w = 2x− y+ z+w = 1

2x+ y− z = 14x+2y+ z = 0

12. Find the solution to the following system in Z5

x+2y+ z−w = 2x− y+ z+w = 0

2x+ y− z = 14x+2y+ z = 3