Uniqueness
Exercise Which of the following consistent systems
have a unique solution (compare with this exercise)?
2x_{1} 

+ 2x_{3} 
+ x_{4} 
= 
2 
x_{1} 
+ x_{2} 
 x_{3} 

= 
1 

x_{2} 
 2x_{3} 
+ x_{4} 
= 
2 
x_{1} 
+ 2x_{2} 
+ x_{3} 
= 
1 
2x_{1} 
+ x_{2} 
+ x_{3} 
= 
1 

3x_{2} 
+ x_{3} 
= 
1 
3x_{1} 

+ x_{3} 
= 
1 
x_{1} 
 x_{2} 
+ x_{3} 

+ 2x_{5} 
= 
1 
2x_{1} 
 2x_{2} 

+ 2x_{4} 
+ 2x_{5} 
= 
0 
 x_{1} 
+ x_{2} 
+ 2x_{3} 
 3x_{4} 
+ x_{5} 
= 
2 
 2x_{1} 
+ 2x_{2} 
+ x_{3} 
 3x_{4} 
 x_{5} 
= 
1 
x_{1} 
 x_{2} 
+ x_{3} 

+ 2x_{5} 
= 
3 
2x_{1} 
 2x_{2} 

+ 2x_{4} 
+ 2x_{5} 
= 
2 
 x_{1} 
+ x_{2} 
+ 2x_{3} 
 3x_{4} 
+ x_{5} 
= 
3 
 2x_{1} 
+ 2x_{2} 
+ x_{3} 
 3x_{4} 
 x_{5} 
= 
0 
x_{1} 
 x_{2} 

= 
1 
 2x_{1} 
+ 3x_{2} 
+ x_{3} 
= 
2 
x_{1} 

+ x_{3} 
= 
5 
x_{1} 
+ 2x_{2} 
+ 3x_{3} 
= 
13 
2x_{1} 
 2x_{2} 
+ 4x_{3} 
= 
5 
Answer
Exercise For what choices of the parameters do
the following systems have a unique solution (compare with this exercise)?
x_{1} 
+ x_{2} 
 x_{3} 
= 
2 
x_{1} 

 ax_{3} 
= 
1 
x_{1} 
+ ax_{2} 

= 
1 
x_{1} 
+ x_{2} 
+ x_{3} 
+ x_{4} 
= 
1 
x_{1} 


+ ax_{4} 
= 
1 

x_{2} 
+ ax_{3} 

= 
1 

v 
+ w 
= 
a 
u 

+ aw 
= 
1 
u 
+ v 

= 
0 

av 
 w 
= 
1 
Answer
Exercise List all shapes of the row echelon forms
of the augmented matrices of consistent systems of 2 equations in 2 variables
with a unique solution. What about systems of other sizes?
Answer
Exercise Let A be a 3 by 3 matrix. Prove that the following are equivalent
 Ax = b is consistent for any b
 Ax = 0 has only the trivial solution
 A can be changed to the identity matrix by the row operations
What about the general case that A is an n by n matrix?
Answer