Question

A scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will have a solution? Discuss.

Short answer

No, the new nonhomogeneous system will not have any solution.

Step-by-step solution

Describe the given statement

It is given that a nonhomogeneous system has ten linear equations with twelve unknowns. There are three free variables in the system. It implies that the system has three non-pivot columns.

Use the rank theorem

Consider the nonhomogeneous system \(Ax = b\), where \(A\) is \(10 \times 12\) matrix. As the system hasthree non-pivot columns, \({\rm{dimNul}}\,A = 3\), and the value of unknown’s \(n\) is 12 . By the rank theorem, \({\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A = n\).

Put the values as shown:

\(\begin{aligned} {\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A &= n\\{\rm{rank}}\,A &= n - {\rm{dim}}\,{\rm{Nul}}\,\,A\\{\rm{rank}}\,A &= 12 - 3\\{\rm{rank}}\,A &= 9\end{aligned}\)

Draw a conclusion

As \({\rm{rank}}\,A\) is 9, \({\rm{col}}\,A\) must be a subspace of \({\mathbb{R}^{10}}\). It means a value of \(b\) exists in \({\mathbb{R}^{10}}\)at which the nonhomogeneous system \(Ax = b\) is inconsistent. Thus, the system \(Ax = b\) may not have a unique solution for all \(b\).