Necessary condition for linear system to have no solution
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 ?
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.
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.
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?

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 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 (righthand 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.