Linear Algebra and Its Applications (5th Edition)

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

Chapter 3 - Determinants - 3.2 Exercises - Page 178: 44

Answer

Let $S(X)=XE$ be a matrix transformation If $S$ is equivalent to one row replacement, then \begin{align*} & \det E=\det I = 1 \\ & \det{AE}=\det{(S(A))}=\det A \\ & \Rightarrow \det{AE}=(\det E)(\det A) \end{align*} If $S$ is equivalent to one row interchange, then \begin{align*} & \det E=-\det I = -1 \\ & \det{AE}=\det{(S(A))}=-\det A \\ & \Rightarrow \det{AE} = (\det E)(\det A) \end{align*} If $S$ is equivalent to one row scaling of $k$, then \begin{align*} & \det E=k\det I = k \\ & \det{AE}=\det{(S(A))}=k\det A \\ & \Rightarrow \det{AE} = (\det E)(\det A) \end{align*}

Work Step by Step

Let $S(X)=XE$ be a matrix transformation If $S$ is equivalent to one row replacement, then \begin{align*} & \det E=\det I = 1 \\ & \det{AE}=\det{(S(A))}=\det A \\ & \Rightarrow \det{AE}=(\det E)(\det A) \end{align*} If $S$ is equivalent to one row interchange, then \begin{align*} & \det E=-\det I = -1 \\ & \det{AE}=\det{(S(A))}=-\det A \\ & \Rightarrow \det{AE} = (\det E)(\det A) \end{align*} If $S$ is equivalent to one row scaling of $k$, then \begin{align*} & \det E=k\det I = k \\ & \det{AE}=\det{(S(A))}=k\det A \\ & \Rightarrow \det{AE} = (\det E)(\det A) \end{align*}
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