Existence
Exercise Which of the following systems are consistent?
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} 
= 
0 

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

+ x_{3} 
= 
0 
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} 
= 
0 
 2x_{1} 
+ 2x_{2} 
+ x_{3} 
 3x_{4} 
 x_{5} 
= 
0 
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 
Answer
Exercise For what choices of the parameters are
the following systems consistent?
x 
+ y 
= 
a 
2x 
+ 2y 
= 
b 
3x 
+ 3y 
= 
c 
4x 
+ 4y 
= 
d 
x_{1} 
+ x_{2} 
+ 2x_{3} 
+ x_{4} 
= 
1 
x_{1} 

+ 2x_{3} 

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

= 
h 

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


u 
+ 2v 
= 
1 
 2x 
+ 2y 
 u 
+ 4v 
= 
h 
x 
 y 
+ u 
 v 
= 
1 
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 Which of the following systems are consistent
for any right side?
2x_{1} 

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

= 
b_{2} 

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

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

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

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

+ 2x_{4} 
+ 2x_{5} 
= 
b_{2} 
 x_{1} 
+ x_{2} 
+ 2x_{3} 
 3x_{4} 
+ x_{5} 
= 
b_{3} 
 2x_{1} 
+ 2x_{2} 
+ x_{3} 
 3x_{4} 
 x_{5} 
= 
b_{4} 
Answer
Exercise For what choices of the parameters are
the following systems consistent for any right side?
x_{1} 
+ x_{2} 
 x_{3} 
= 
b_{1} 
x_{1} 

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

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


+ ax_{4} 
= 
b_{2} 

x_{2} 
+ ax_{3} 

= 
b_{3} 

v 
+ w 
= 
b_{1} 
u 

+ aw 
= 
b_{2} 
u 
+ v 

= 
b_{3} 

av 
 w 
= 
b_{4} 
Answer
Exercise List all shapes of the row echelon forms of
the augmented matrices of consistent systems of 2 equations and 2 variables.
What about systems of other sizes?
Answer
Exercise List all shapes of the row echelon forms
of 2 by 2 matrices A so that Ax = b
has solutions for any b. What about matrices of other sizes?
Answer
Exercise Without any computation, determine the
consistency of the systems.
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} 
= 
1 
2x_{1} 
+ 4x_{2} 
= 
2 
x_{1} 
+ 2x_{2} 
= 
a 
 x_{1} 
 2x_{2} 
= 
b 
x_{1} 
+ x_{2} 
+ x_{3} 
= 
1 
x_{1} 
+ 2x_{2} 
+ 3x_{3} 
= 
2 
2x_{1} 
+ 3x_{2} 
+ 4x_{3} 
= 
0 
Answer
Exercise Show that the system Ax
= 0, with the zero vector on the right side, is always consistent.
Answer