Question

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? Discuss.

Short answer

No, it is not possible to find two nonzero solutions of the associated homogeneous system that are not multiples of each other.

Step-by-step solution

Describe the given statement

It is given that a nonhomogeneous system has nine linear equations with ten unknowns. A solution exists for all the possible constants on the right side of the equations. This implies that the maximum rank of matrix formed from the nonhomogeneous system is 9 as it has 9 pivot places.

Use the rank theorem

The value of \(n\) of the unknown is 10, and the rank is 9. By the rank theorem,\({\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A = n\).

Put the values as shown below:

\(\begin{aligned} {\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A &= n\\9 + {\rm{dim}}\,{\rm{Nul}}\,\,A &= 10\\{\rm{dim}}\,{\rm{Nul}}\,\,A &= 10 - 9\\{\rm{dim}}\,{\rm{Nul}}\,\,A &= 1\end{aligned}\)

Draw a conclusion

As the value of \({\rm{dim}}\,{\rm{Nul }}A\) is 1, the number of non-pivot columns is 1. Thus, it is not possible to find two nonzero solutions of the associated homogeneous system.