Question

Is it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side? Explain.

Short answer

Yes,it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants.

No, it is not possible for such a system to have a unique solution for every right-hand side.

Step-by-step solution

Describe the given statement

It is given that a nonhomogeneous system has seven linear equations with six unknowns. The system has unique solutions for some right-hand side of constants. This implies that the system has at most six pivot positions.

Use the rank theorem

Consider the nonhomogeneous system \(Ax = b\), where \(A\) is \(7 \times 6\) matrix. As the system hasat most six pivot positions, \({\rm{rank}}\,A \le 6\), and the value of unknown’s \(n\) is 6 . By the rank theorem, \({\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A = n\).

Put the values as shown:

\(\begin{aligned} {\rm{rank}}\,A + {\rm{dim}}\,{\rm{Nul}}\,\,A &= n\\{\rm{dim}}\,{\rm{Nul}}\,\,A &= n - {\rm{rank}}\,A\\{\rm{dim}}\,{\rm{Nul}}\,\,A &\ge 6 - 6\\{\rm{dim}}\,{\rm{Nul}}\,\,A &\ge 0\end{aligned}\)

Draw a conclusion

If \({\rm{dim}}\,{\rm{Nul}}\,\,A = 0\), the system \(Ax = b\) has no free variable and its solution is unique. The value of \({\rm{dimcol}}\,A\) is also 6. Moreover, \({\rm{col}}\,A\) is a subspace of \({\mathbb{R}^7}\) as \({\rm{rank}}\,A \le 6\). So, a value of \(b\) must exist in \({\mathbb{R}^7}\)at which the nonhomogeneous system \(Ax = b\) is inconsistent. Thus, the system \(Ax = b\) may not have a unique solution for all \(b\).