Question

Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations’ right sides to make the new system inconsistent? Explain.

Short answer

No, by using the rank theorem and the invertible matrix theorem.

Step-by-step solution

Describe the given statement

Consider the nonhomogeneous system \(Ax = b\), where A is the \(6 \times 8\) matrix. From the given statement, \({\rm{dim Null }}A = 2\).

Use the rank theorem

Bythe rank theorem, you get

\(\begin{aligned} {\rm{rank }}A &= n - {\rm{dim Null }}A\\ &= 8 - 2\\{\rm{rank }}A &= 6.\end{aligned}\)

As \({\rm{dim Col }}A = {\rm{rank }}A\), \({\rm{dim Col }}A = 6\). Since Col A is the subspace of \({\mathbb{R}^6}\), \({\rm{Col }}A = {\mathbb{R}^6}\).

Draw a conclusion

By the invertible matrix theorem, for everyb in \({\mathbb{R}^6}\), the system \(Ax = b\) has a unique solution. Hence, it is impossible to change the entries in b to convert \(Ax = b\) into an inconsistent system.