# solutions of a homogeneous system

• Nov 9th 2009, 01:42 PM
Projectt
solutions of a homogeneous system
could someone help me with this problem?

"suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. is it possible to find two nonzero solutions of the associated homogeneous system that are NOT multiples of each other?"
• Nov 10th 2009, 05:11 AM
HallsofIvy
Quote:

Originally Posted by Projectt
could someone help me with this problem?

"suppose a nonhomogeneous system of nine linear equations in ten unknowns has a solution for all possible constants on the right sides of the equations. is it possible to find two nonzero solutions of the associated homogeneous system that are NOT multiples of each other?"

One solution of any homogeneous system is the "0" solution: all unknowns equal to 0. If there exist other solutions, then the determinant of the coefficient matrix is not 0 (equivalently, the coefficient matrix is not invertible). That means that, if we were to row reduce the coefficient matrix, we would have at least one row of all "0"s. Certainly, it would be possible to have a "right side" such that the last column of that row, after row reducing the augmented matrix, would NOT be 0.