Existence
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 By the row operations in an earlier
exercise,
we have the row echelon forms of the coefficient matrices (obtained by dropping the last columns).
[ 
1 
1 
1 
0 
] 
0 
1 
2 
1 
0 
0 
0 
3 
[ 
1 
2 
1 
] 
0 
3 
1 
0 
0 
0 
0 
0 
0 
[ 
1 
1 
1 
0 
2 
] 
0 
0 
1 
1 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
Since there is no zero row in the first row echelon form, the first system
is consistent for any right side. Since there are zero rows in the second and
the third row echelon forms, the second and the third systems are not always
consistent.