## Necessary condition for linear system to have no solution

Forum for the GRE subject test in mathematics.
mathQ
Posts: 41
Joined: Thu Mar 25, 2010 12:14 am

### Necessary condition for linear system to have no solution

Is the Necessary condition for linear system to have no solution is det of coeff. matrix = 0 ?

origin415
Posts: 61
Joined: Fri Oct 23, 2009 11:42 pm

### Re: Necessary condition for linear system to have no solution

If the determinant is nonzero than there exists exactly one solution. If the determinant is zero, there could be no solutions, or there could be infinitely many. It just means the matrix isn't invertible.

As a trivial example, the system of equations
x + 0y = 0
0x + 0y = 0
has a coefficient matrix of
1 0
0 0
Which has determinant zero. However, there are infinite solutions, x = 0 and y is anything.
Changing this, you could have
x + 0y = 0
0x + 0y = 1
Which is obviously absurd, so has no solutions.

mathQ
Posts: 41
Joined: Thu Mar 25, 2010 12:14 am

### Re: Necessary condition for linear system to have no solution

origin415 wrote:If the determinant is nonzero than there exists exactly one solution. If the determinant is zero, there could be no solutions, or there could be infinitely many. It just means the matrix isn't invertible.

As a trivial example, the system of equations
x + 0y = 0
0x + 0y = 0
has a coefficient matrix of
1 0
0 0
Which has determinant zero. However, there are infinite solutions, x = 0 and y is anything.
Changing this, you could have
x + 0y = 0
0x + 0y = 1
Which is obviously absurd, so has no solutions.

Thanks Origin for the nice explanation.

So, if I am given a question with a linear system for e.g.

x + y + z=0
x + 2y + 3z = 0
x + 3y + bz =0

Does there exists a value of b for which system has no solution?

EugeneKudashev
Posts: 27
Joined: Tue Apr 06, 2010 8:22 am

### Re: Necessary condition for linear system to have no solution

I may be mistaken, but: your system is homogeneous (right-hand side is zero vector). such a system (as I feel) always has a trivial solution - zero vector. thus there is no such value of b.