Existence
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 The coefficients of the systems have appeared
in an earlier exercise. By
the row operations there, we have the row echelon forms of the coefficient matrices
(obtained by dropping the last columns).
[ 
1 
1 
1 
] 
0 
1 
1  a 
0 
0 
2a  a^{2} 
[ 
1 
1 
1 
1 
] 
0 
1 
1 
1  a 
0 
0 
a  1 
a  1 
[ 
1 
0 
a 
] 
0 
1 
1 
0 
0 
 a  1 
0 
0 
0 
The condition for the first system to be always consistent is 2a  a^{2}
≠ 0, which is equivalent to a ≠ 0 and 2. The condition for the
second system to be always consistent is a ≠ 1. Since [row 4] in the third row echelon form is a zero row,
the third system is not always consistent for any choice of a.