When is a 3x3 linear system consistent for arbitrary right-hand sides?

[hkus10]

1) Find an equation relating a, b, and c so that the linear system
2x+2y+3z = a
3x- y+5z = b
x-3y+2z = c
is consistent for any values of a, b, and c that satisfy that equation.
what is the method to solve this problem?

2) In the following linear system, determine all values of a for which the resulting linear system has
a) no solution;
b) a unique solution;
c) infinitely many solutions:
x + y - z = 2
x + 2y + z = 3
x + y + (a^2 - 5)z = a

For these two questions:
Do I make this to be a reduced echelon form first?
If yes, how to make it with some variables a, b, and c?
If no, what is the right approach for this problem?

Thanks

 

[chiro]

1) Find an equation relating a, b, and c so that the linear system
2x+2y+3z = a
3x- y+5z = b
x-3y+2z = c
is consistent for any values of a, b, and c that satisfy that equation.
what is the method to solve this problem?

2) In the following linear system, determine all values of a for which the resulting linear system has
a) no solution;
b) a unique solution;
c) infinitely many solutions:
x + y - z = 2
x + 2y + z = 3
x + y + (a^2 - 5)z = a

For these two questions:
Do I make this to be a reduced echelon form first?
If yes, how to make it with some variables a, b, and c?
If no, what is the right approach for this problem?

Thanks

Hello hkus10

Putting the solution in reduced-echelon form is a good start since it can be understood from first principles.

If you wanted to do it this way, you basically form the augmented matrix and for each row operation you do on the system matrix, you do exactly the same thing in the augmented row matrix (ie [a b c]^T)

Another way is to use the determinant to verify if the system is non-singular. If a system is singular then there is no unique solution.

Are you aware of the determinant, its use and meaning?