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Expert-verified Found in: Page 191 ### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384 # 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.

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.

See the step by step solution

## Step 1: 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.

## Step 2: Use the rank theorem

Consider the nonhomogeneous system $$Ax = b$$, where $$A$$ is $$7 \times 6$$ matrix. As the system has at 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}

## Step 3: 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 $$b$$. ### Want to see more solutions like these? 