Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - 1.4 Exercises - Page 42: 31

Answer

A$\vec x$=$\vec b$ cannot be consistent for all vectors b in $\mathbb{R}^{3}$. For any mxn matrix A with m>n, A$\vec x$=$\vec b$ cannot be consistent for all vectors $\vec b$ in $\mathbb{R}^{m}$.

Work Step by Step

The matrix A is given to be a 3x2 matrix, so it can have at most 2 pivot positions because it has only two columns. Thus, it cannot have a pivot position in every row. By Theorem 4, A$\vec x$=$\vec b$ cannot be consistent for all vectors b in $\mathbb{R}^{3}$. In general, for any mxn matrix A with m>n, A$\vec x$=$\vec b$ cannot be consistent for all vectors $\vec b$ in $\mathbb{R}^{m}$.
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