What is condition for non trivial solutions of a system of linear equations?

Answer 1

Actually, having a zero determinant is what you want.....contrary to the above answer.

Trivial solutions are brought up in talking about solutions to homogenous systems of equations, ie:

Ax = 0

For matrix A, your vector of variables, and a zero vector.

The trivial solution is simply where x is also a vector of zeros. Obviously, one could multiply an mxn matrix by a nx1 vector of zeros to obtain a zero vector, but this is trivial, eh?

****A homogeneous system has a non-trivial solution if and only if the system has at least one free variable. **** This follows from the Existence and Uniqueness Thrm.


But of course, if you've got a free variable, you've got a row of zeros, implying that you have a zero determinant, which is why the above answer is wrong. To further see this, assume that A is non-singular. Then the equation, where 0 is the zero vector,

Ax=0

Has the solution

x = A^(-1)0

But of course A^(-1)0 = a zero vector, producing the trivial result.

Answer 2

Trivial Solution

Answer 3

I have no idea hat you mean with "non trivial solutions".
If you mean a system of homogene linear equotions the trivial solution is the 0-vector.

A system of lineair equations has exactly ONE solution if the determinant of the system is not 0.

If the determinant is equal zero
then some equations are lineair dependent and you will have an infinite number of solutions or NO solutions at all.

Answer 4

Determinants

Answer 5

you can use the theory of determinants to answer this question.

Answer 6

[-b=/-(b^2-4ac)^1/2]/2a should not be zero