91Ó°ÊÓ

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.

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:

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

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 \(b\).