Section 1.1 Matrices and Systems of Equations
Exercise 6 Solve the following linear systems.
 3 x1  2 x2  x3  0

(f)  2 x1  x2  x3  2
2 x  x  2 x  1
3
 1 2
2
1
 3 x1  3 x2  2 x3  1

3
3

(g)  x1  2 x2  x3 
2
2

12
1
1
 2 x1  2 x2  5 x3  10

x2  x3  x4  0

 3x  3x  4 x  7
 1
3
4
(h) 
 x1  x2  x3  2 x4  6
2 x1  3 x2  x3  3 x4  6
 m1 x1  x2  b1
, where m1 , m2 , b1 and b2 are constants:
m2 x1  x2  b2
9. Given a system of the form 
(a) Show that the system will have a unique solution if m1  m2 .
(b) Show that if m1  m2 , then the system will consistent only if b1  b2 .
(c) Give a geometric interpretation of parts(a) and (b).
 a11 x1  a12 x2  0
, where a11 , a12 , a21 and a22 are
a21 x1  a22 x2  0
10. Consider a system of the form 
constants. Explain why a system of this form must be consistent.
Section1.2 Row Echelon Form
Exercise 5. Solve the following linear systems
  x1  2 x2  x3  2
 2 x  2 x  x  4

1
2
3
(i) 
 3 x1  2 x2  2 x3  5
3 x1  8 x2  5 x3  17
 x1  2 x2  3x3  x4  1

(j)   x1  x2  4 x3  x4  6
2 x  4 x  7 x  x  1
1
2
3
4

 x1  3x2  x3  x4  3

(k) 2 x1  2 x2  x3  2 x4  8
 x  5x 
x4  5
2
 1
 x1  3 x2  x3  1
 2x  x  x  2
 1 2 3
(l) 
 x1  4 x2  2 x3  1
5 x1  8 x2  2 x3  5
 1 2 1 0


9. Consider a linear system whose augmented matrix is of the form  2 5 3 0 
 1 1  0 


(a) Is it possible for the system to be inconsistent? Explain.
(b) For what values of
 will the system have infinitely many solutions?
1 1 3 2 


10. Consider a linear system whose augmented matrix is of the form 1 2 4 3 
1 3 a b 


(a) For what values of a and b will the system have infinitely many solutions?
(b) For what values of a and b will the system be inconsistent?